%------------------------------------------------------------------------------
% File     : ITP259^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Delete 01385_095048
% 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    : 0072_VEBT_Delete_01385_095048 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11292 (5332 unt;1120 typ;   0 def)
%            Number of atoms       : 29943 (12421 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 118901 (2752   ~; 508   |;1914   &;102265   @)
%                                         (   0 <=>;11462  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   87 (  86 usr)
%            Number of type conns  : 4744 (4744   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1037 (1034 usr;  77 con; 0-8 aty)
%            Number of variables   : 26169 (2350   ^;23060   !; 759   ?;26169   :)
% 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
%            2022-02-18 06:37:04.880
%------------------------------------------------------------------------------
% Could-be-implicit typings (86)
thf(ty_n_t__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,
    produc1319942482725812455at_nat: $tType ).

thf(ty_n_t__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,type,
    set_Pr4329608150637261639at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc4471711990508489141at_nat: $tType ).

thf(ty_n_t__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,
    produc3843707927480180839at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr8693737435421807431at_nat: $tType ).

thf(ty_n_t__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,
    produc859450856879609959at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
    set_fi4554929511873752355omplex: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7248412053542808358at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
    set_fi7789364187291644575l_real: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    filter6041513312241820739omplex: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_li5450038453877631591at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8923325533196201883nteger: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    produc7272778201969148633d_enat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    filter2146258269922977983l_real: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    option4927543243414619207at_nat: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    filter1242075044329608583at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    set_Pr6218003697084177305l_real: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    list_P5707943133018811711nt_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    produc9072475918466114483BT_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    produc4411394909380815293omplex: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    set_list_VEBT_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    produc334124729049499915VEBT_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    produc2504756804600209347T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc6271795597528267376eger_o: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Extended____Nat__Oenat_J_J,type,
    set_li5464603477888414924d_enat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    produc2422161461964618553l_real: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    product_prod_int_int: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
    set_list_complex: $tType ).

thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    list_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
    set_list_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    product_prod_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    product_prod_o_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    list_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    set_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
    list_Extended_enat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    set_Extended_enat: $tType ).

thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
    list_complex: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
    set_list_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

thf(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

thf(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

thf(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

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__String__Oliteral,type,
    literal: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $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,
    int: $tType ).

% Explicit typings (1034)
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_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
    bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_Eo_J_001_062_It__Real__Oreal_M_Eo_J,type,
    bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
    bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
    bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
    bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: complex > nat > complex ).

thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > nat > int ).

thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > nat > rat ).

thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: real > nat > real ).

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_Oand__not__num,type,
    bit_and_not_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > int > int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

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__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

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__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

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,
    bit_take_bit_num: nat > num > option_num ).

thf(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

thf(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

thf(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
    comple4887499456419720421f_real: set_real > real ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
    comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).

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__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

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_Ocnj,type,
    cnj: complex > complex ).

thf(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > real > complex ).

thf(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

thf(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

thf(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Complex_Orcis,type,
    rcis: real > real > complex ).

thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

thf(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > int > int ).

thf(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
    euclid3395696857347342551nt_int: int > int ).

thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
    euclid3398187327856392827nt_nat: nat > nat ).

thf(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
    extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
    extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nat__Oenat,type,
    extend5688581933313929465d_enat: extended_enat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
    comm_s8582702949713902594nteger: code_integer > nat > code_integer ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
    semiri3624122377584611663nteger: nat > code_integer ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
    filtermap_real_real: ( real > real ) > filter_real > filter_real ).

thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).

thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).

thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
    finite_card_list_nat: set_list_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
    finite_finite_o: set_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
    finite4001608067531595151d_enat: set_Extended_enat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
    finite_finite_list_o: set_list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    finite1862508098717546133d_enat: set_li5464603477888414924d_enat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
    finite_finite_num: set_num > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
    finite_finite_rat: set_rat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
    finite_finite_real: set_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Nat__Onat,type,
    bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).

thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
    bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    bij_be6293887246118711976st_nat: ( nat > list_nat ) > set_nat > set_list_nat > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bij_be8693218025023041337at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Obij__betw_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bij_be5333170631980326235at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
    comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).

thf(sy_c_Fun_Oid_001_Eo,type,
    id_o: $o > $o ).

thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
    inj_on_int_nat: ( int > nat ) > set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
    inj_on_nat_int: ( nat > int ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on5538052773655684606at_nat: ( nat > product_prod_nat_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: ( nat > char ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on2178005380612969504at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).

thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).

thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).

thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).

thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).

thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).

thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
    map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).

thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
    strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(sy_c_Fun__Def_Ois__measure_001t__Int__Oint,type,
    fun_is_measure_int: ( int > nat ) > $o ).

thf(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set_Pr4329608150637261639at_nat ).

thf(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Oreduction__pair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).

thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
    gcd_Gcd_int: set_int > int ).

thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(sy_c_GCD_OGcd__class_OLcm_001t__Int__Oint,type,
    gcd_Lcm_int: set_int > int ).

thf(sy_c_GCD_OGcd__class_OLcm_001t__Nat__Onat,type,
    gcd_Lcm_nat: set_nat > nat ).

thf(sy_c_GCD_Obezw,type,
    bezw: nat > nat > product_prod_int_int ).

thf(sy_c_GCD_Obezw__rel,type,
    bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
    gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).

thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
    gcd_gcd_int: int > int > int ).

thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(sy_c_GCD_Ogcd__class_Olcm_001t__Int__Oint,type,
    gcd_lcm_int: int > int > int ).

thf(sy_c_GCD_Ogcd__class_Olcm_001t__Nat__Onat,type,
    gcd_lcm_nat: nat > nat > nat ).

thf(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
    abs_abs_Code_integer: code_integer > code_integer ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
    abs_abs_complex: complex > complex ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: rat > rat ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: complex > complex > complex ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
    minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > rat > rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
    monoid_nat: ( nat > nat > nat ) > nat > $o ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
    one_one_Code_integer: code_integer ).

thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
    one_one_complex: complex ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
    plus_plus_complex: complex > complex > complex ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > num > num ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: rat > rat > rat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: real > real > real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__String__Oliteral,type,
    plus_plus_literal: literal > literal > literal ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
    sgn_sgn_complex: complex > complex ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
    sgn_sgn_int: int > int ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
    sgn_sgn_real: real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
    times_times_complex: complex > complex > complex ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: num > num > num ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
    times_times_rat: rat > rat > rat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
    uminus_uminus_real: real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
    zero_zero_complex: complex ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
    zero_zero_rat: rat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__String__Oliteral,type,
    zero_zero_literal: literal ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
    groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Code____Numeral__Ointeger,type,
    groups6621422865394947399nteger: ( complex > code_integer ) > set_complex > code_integer ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
    groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
    groups2025484359314973016at_int: ( extended_enat > int ) > set_Extended_enat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
    groups2027974829824023292at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    groups1392844769737527556at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    groups4148127829035722712t_real: ( extended_enat > real ) > set_Extended_enat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
    groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
    groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Rat__Orat,type,
    groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
    groups8778361861064173332t_real: ( int > real ) > set_int > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
    groups7108830773950497114d_enat: ( nat > extended_enat ) > set_nat > extended_enat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
    groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
    groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
    groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
    groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
    groups1932886352136224148al_int: ( real > int ) > set_real > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
    groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
    groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
    groups8097168146408367636l_real: ( real > real ) > set_real > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
    groups5107569545109728110t_real: ( set_nat > real ) > set_set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    groups771621172384141258BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Int__Oint,type,
    groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    groups766887009212190081x_real: ( complex > real ) > set_complex > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Complex__Ocomplex,type,
    groups4622424608036095791omplex: ( extended_enat > complex ) > set_Extended_enat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Int__Oint,type,
    groups2878480467620962989at_int: ( extended_enat > int ) > set_Extended_enat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
    groups2880970938130013265at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    groups2245840878043517529at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    groups97031904164794029t_real: ( extended_enat > real ) > set_Extended_enat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
    groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Nat__Onat,type,
    groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
    groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
    groups2316167850115554303t_real: ( int > real ) > set_int > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
    groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
    groups129246275422532515t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    groups4077766827762148844at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
    groups6036352826371341000t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    groups713298508707869441omplex: ( real > complex ) > set_real > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
    groups4694064378042380927al_int: ( real > int ) > set_real > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
    groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
    groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
    groups1681761925125756287l_real: ( real > real ) > set_real > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
    groups3619160379726066777t_real: ( set_nat > real ) > set_set_nat > real ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
    groups4561878855575611511st_nat: list_nat > nat ).

thf(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    nO_MAT8252062027627875367nteger: code_integer > code_integer > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    nO_MAT8947977539597988553omplex: complex > complex > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    nO_MAT9165723751696580935x_real: complex > real > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Int__Oint,type,
    nO_MATCH_int_int: int > int > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Nat__Onat,type,
    nO_MATCH_int_nat: int > nat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Rat__Orat,type,
    nO_MATCH_int_rat: int > rat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Real__Oreal,type,
    nO_MATCH_int_real: int > real > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Int__Oint,type,
    nO_MATCH_nat_int: nat > int > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Nat__Onat,type,
    nO_MATCH_nat_nat: nat > nat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Rat__Orat,type,
    nO_MATCH_nat_rat: nat > rat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Real__Oreal,type,
    nO_MATCH_nat_real: nat > real > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Rat__Orat,type,
    nO_MATCH_rat_rat: rat > rat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Real__Oreal_001t__Int__Oint,type,
    nO_MATCH_real_int: real > int > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Real__Oreal_001t__Rat__Orat,type,
    nO_MATCH_real_rat: real > rat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Real__Oreal_001t__Real__Oreal,type,
    nO_MATCH_real_real: real > real > $o ).

thf(sy_c_HOL_OThe_001t__Int__Oint,type,
    the_int: ( int > $o ) > int ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).

thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
    the_real: ( real > $o ) > real ).

thf(sy_c_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).

thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(sy_c_If_001t__Complex__Ocomplex,type,
    if_complex: $o > complex > complex > complex ).

thf(sy_c_If_001t__Extended____Nat__Oenat,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Num__Onum,type,
    if_num: $o > num > num > num ).

thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
    infini8530281810654367211te_nat: set_nat > nat > nat ).

thf(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(sy_c_Int_Ocr__int,type,
    cr_int: product_prod_nat_nat > int > $o ).

thf(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Ointrel,type,
    intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Int_Onat,type,
    nat2: int > nat ).

thf(sy_c_Int_Opcr__int,type,
    pcr_int: product_prod_nat_nat > int > $o ).

thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
    power_int_real: real > int > real ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
    ring_11222124179247155820nteger: set_Code_integer ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
    ring_1_Ints_complex: set_complex ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
    ring_1_Ints_int: set_int ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
    ring_1_Ints_rat: set_rat ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: int > int ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
    ring_1_of_int_rat: int > rat ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
    ring_1_of_int_real: int > real ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
    inf_inf_int: int > int > int ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Osemilattice__neutr_001t__Nat__Onat,type,
    semila9081495762789891438tr_nat: ( nat > nat > nat ) > nat > $o ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: int > int > int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__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,type,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Extended____Nat__Oenat,type,
    lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
    lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Num__Onum,type,
    lattic402713867396545063at_num: ( extended_enat > num ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    lattic3210252021154270693at_rat: ( extended_enat > rat ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    lattic1189837152898106425t_real: ( extended_enat > real ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Num__Onum,type,
    lattic5003618458639192673nt_num: ( int > num ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
    lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
    lattic2675449441010098035t_real: ( int > real ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__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__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_Lifting_OQuotient_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    quotie3684837364556693515t_real: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > real ) > ( real > nat > rat ) > ( ( nat > rat ) > real > $o ) > $o ).

thf(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    quotie1194848508323700631at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int > $o ) > $o ).

thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( nat > real ) > filter_nat > $o ).

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__Extended____Nat__Oenat,type,
    count_101369445342291426d_enat: list_Extended_enat > extended_enat > 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 ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    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_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_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > 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__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_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
    set_Extended_enat2: list_Extended_enat > set_Extended_enat ).

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 ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

thf(sy_c_List_Olist_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 ).

thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    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__Complex__Ocomplex,type,
    list_update_complex: list_complex > nat > complex > list_complex ).

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 ).

thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
    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 ).

thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > nat > $o ).

thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
    nth_complex: list_complex > nat > complex ).

thf(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > nat > int ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    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 ).

thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    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__Nat__Onat,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: nat > $o > list_o ).

thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
    replicate_complex: nat > complex > list_complex ).

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 ).

thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    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 ).

thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

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_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

thf(sy_c_List_Oupto__aux,type,
    upto_aux: int > int > list_int > list_int ).

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__Complex__Ocomplex,type,
    semiri3842193898606819883omplex: set_complex ).

thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: nat > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    size_s3941691890525107288d_enat: list_Extended_enat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: nat > int ).

thf(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: int > nat ).

thf(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: nat > nat > $o ).

thf(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

thf(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

thf(sy_c_NthRoot_Oroot,type,
    root: nat > real > real ).

thf(sy_c_NthRoot_Osqrt,type,
    sqrt: real > real ).

thf(sy_c_Num_OBitM,type,
    bitM: num > num ).

thf(sy_c_Num_Oinc,type,
    inc: num > num ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > num > int ).

thf(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

thf(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: num > int ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

thf(sy_c_Num_Opow,type,
    pow: num > num > num ).

thf(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

thf(sy_c_Num_Osqr,type,
    sqr: num > num ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: int > ( num > int ) > option_num > int ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__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,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
    ord_Least_real: ( real > $o ) > real ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
    ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_eq_set_num: set_num > set_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > int > int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > num > num ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: rat > rat > rat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > real > real ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_ma4205026669011143323d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: set_int > set_int > set_int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: set_real > set_real > set_real ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: int > int > int ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: ( nat > real ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > nat > complex ).

thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).

thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > nat > int ).

thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > nat > nat ).

thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > nat > rat ).

thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: real > nat > real ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).

thf(sy_c_Product__Type_OPair_001t__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_001t__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,type,
    produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
    produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: product_prod_int_int > rat ).

thf(sy_c_Rat_OFract,type,
    fract: int > int > rat ).

thf(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

thf(sy_c_Rat_ORep__Rat,type,
    rep_Rat: rat > product_prod_int_int ).

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: rat > real ).

thf(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(sy_c_Rat_Opcr__rat,type,
    pcr_rat: product_prod_int_int > rat > $o ).

thf(sy_c_Rat_Opositive,type,
    positive: rat > $o ).

thf(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

thf(sy_c_Rat_Oratrel,type,
    ratrel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Real_OReal,type,
    real2: ( nat > rat ) > real ).

thf(sy_c_Real_Ocauchy,type,
    cauchy: ( nat > rat ) > $o ).

thf(sy_c_Real_Ocr__real,type,
    cr_real: ( nat > rat ) > real > $o ).

thf(sy_c_Real_Opcr__real,type,
    pcr_real: ( nat > rat ) > real > $o ).

thf(sy_c_Real_Opositive,type,
    positive2: real > $o ).

thf(sy_c_Real_Orealrel,type,
    realrel: ( nat > rat ) > ( nat > rat ) > $o ).

thf(sy_c_Real_Orep__real,type,
    rep_real: real > nat > rat ).

thf(sy_c_Real_Ovanishes,type,
    vanishes: ( nat > rat ) > $o ).

thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: real > real > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: complex > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: real > real ).

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > real > real ).

thf(sy_c_Relation_OId_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    id_Pro2258643101195443293at_nat: set_Pr8693737435421807431at_nat ).

thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).

thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: int > int > $o ).

thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: nat > nat > $o ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

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__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

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__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

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__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_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__Complex__Ocomplex,type,
    summable_complex: ( nat > complex ) > $o ).

thf(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: ( nat > int ) > $o ).

thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: ( nat > nat ) > $o ).

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__Extended____Nat__Oenat,type,
    collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).

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__Extended____Nat__Oenat_J,type,
    collec8433460942617342167d_enat: ( list_Extended_enat > $o ) > set_li5464603477888414924d_enat ).

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__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__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_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__List__Olist_It__Nat__Onat_J,type,
    image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

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_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: int > set_int > set_int ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

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__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

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__Extended____Nat__Oenat,type,
    set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat ).

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__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__Extended____Nat__Oenat,type,
    set_or8332593352340944941d_enat: extended_enat > set_Extended_enat ).

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__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__Extended____Nat__Oenat,type,
    set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).

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_String_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: literal > ( product_unit > real ) > real ).

thf(sy_c_String_OLiteral,type,
    literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).

thf(sy_c_String_Ochar_OChar,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: char > nat ).

thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: nat > char ).

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__Int__Oint,type,
    topolo4899668324122417113eq_int: ( nat > int ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: ( nat > num ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: ( nat > set_nat ) > $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_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: ( nat > real ) > $o ).

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__Complex__Ocomplex,type,
    cos_complex: complex > complex ).

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_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: ( nat > complex ) > nat > complex ).

thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: ( nat > int ) > nat > int ).

thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: ( nat > rat ) > nat > rat ).

thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: ( nat > real ) > nat > real ).

thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: complex > complex ).

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_Opowr__real,type,
    powr_real2: real > real > real ).

thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: complex > complex ).

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__Complex__Ocomplex,type,
    tan_complex: complex > complex ).

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_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bi_tot896582865486249351at_int: ( product_prod_nat_nat > int > $o ) > $o ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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_Oelim__dead,type,
    vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $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__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_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_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > 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_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_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__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $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_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $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_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_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__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $o ).

thf(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > set_complex > $o ).

thf(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > set_Extended_enat > $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__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__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__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__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 ).

% Relevant facts (10129)
thf(fact_0__C5_OIH_C_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ X ) @ Y )
      = ( ( X != Y )
        & ( vEBT_V8194947554948674370ptions @ summary @ Y ) ) ) ).

% "5.IH"(2)
thf(fact_1_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_2__C5_Ohyps_C_I3_J,axiom,
    ( m
    = ( suc @ na ) ) ).

% "5.hyps"(3)
thf(fact_3__C5_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "5.hyps"(1)
thf(fact_4__C5_Ohyps_C_I7_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5.hyps"(7)
thf(fact_5__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).

% "5.hyps"(6)
thf(fact_6_mimapr,axiom,
    ord_less_nat @ mi @ ma ).

% mimapr
thf(fact_7_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_8__C5_Ohyps_C_I4_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "5.hyps"(4)
thf(fact_9_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X3: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
          | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).

% both_member_options_def
thf(fact_10_False,axiom,
    xa = mi ).

% False
thf(fact_11__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
    ( ( ord_less_eq_nat @ mi @ xa )
    & ( ord_less_eq_nat @ xa @ ma ) ) ).

% \<open>mi \<le> x \<and> x \<le> ma\<close>
thf(fact_12_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] :
          ( X
         != ( plus_plus_nat @ N @ N ) )
     => ~ ! [N: nat] :
            ( X
           != ( plus_plus_nat @ N @ ( suc @ N ) ) ) ) ).

% even_odd_cases
thf(fact_13_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).

% max_in_set_def
thf(fact_14_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% min_in_set_def
thf(fact_15__C5_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X2 @ na )
        & ! [Xa: nat,Xb: nat] :
            ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X2 @ Xa ) @ Xb )
            = ( ( Xa != Xb )
              & ( vEBT_V8194947554948674370ptions @ X2 @ Xb ) ) ) ) ) ).

% "5.IH"(1)
thf(fact_16__092_060open_062x_A_061_Ami_092_060close_062,axiom,
    xa = mi ).

% \<open>x = mi\<close>
thf(fact_17_nmpr,axiom,
    ( ( ord_less_eq_nat @ one_one_nat @ na )
    & ( m
      = ( suc @ na ) ) ) ).

% nmpr
thf(fact_18_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_19_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_20_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_21_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_22_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_23_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_24_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_25_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_26_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_27_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_28_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_29_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_30_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_31_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_32_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_33_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_34_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_35_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_36_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_37_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_38_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_39_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_40_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_41_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_42_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_43_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_44_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_45_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_46_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_47_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_48_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_49_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_50_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_51_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_52_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_53_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_54_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_55_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_56_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_57_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_58_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_59_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_60_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_61_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_62_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_63_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_64_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_65_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_66_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_67_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_68_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_69_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_70_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_71_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_72_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_73_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_74_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_75_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A3: set_complex] :
      ( ( collect_complex
        @ ^ [X3: complex] : ( member_complex @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A3: set_real] :
      ( ( collect_real
        @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X3: int] : ( member_int @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_87_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X4: real] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_88_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X4: list_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_89_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X4: set_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_90_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_91_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_92_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_93_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_94_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_95_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_96_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_97_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_98_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_99_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_100_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_101_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_102_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B3 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_103_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_104_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_105_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_106_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A3 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_107_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_108_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A3 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_109_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_110_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_111_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_112_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_113_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_114_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_115_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_116_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_117_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_118_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_119_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_120_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_121_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( P @ N )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
              & ~ ( P @ M2 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_122_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_123_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_124_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_125_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_126_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_127_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_128_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_129_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_130_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_131_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_132_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_133_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_134_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_135_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_136_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_137_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_138_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_139_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_140_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_141_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_142_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C2: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_143_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_144_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_145_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_146_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_147_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_148_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_149_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_150_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_151_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_152_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_153_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_154_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_155_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_156_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_157_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_158_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_159_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_160_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_161_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_162_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_163_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_164_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_165_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_168_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_169_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_170_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_171_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_172_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_173_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_174_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_175_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_176_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_177_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_178_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_179_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_180_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_181_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_182_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_183_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_184_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_185_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_186_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_187_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_188_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_189_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_190_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_191_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_192_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_193_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_194_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_195_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_196_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_197_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_198_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_199_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_200_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_201_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_202_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_203_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_204_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_205_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_206_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M3: nat] :
            ( ( M
              = ( suc @ M3 ) )
            & ( ord_less_nat @ N2 @ M3 ) ) ) ) ).

% Suc_less_eq2
thf(fact_207_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N2 )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_208_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_209_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_210_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N2 )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_211_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_212_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_213_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_214_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_215_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_216_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_217_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z: nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
           => ( R @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_218_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ M @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_219_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_220_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_221_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_222_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_223_Suc__le__D,axiom,
    ! [N2: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_224_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_225_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_226_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_227_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_228_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_229_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A3 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_230_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_231_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_232_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_233_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N3: nat] :
          ( ( ord_less_nat @ M6 @ N3 )
          | ( M6 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_234_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_235_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M6 @ N3 )
          & ( M6 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_236_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_237_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_238_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_239_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_240_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_241_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_242_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_243_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_244_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_245_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_246_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_247_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_248_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_249_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N: nat] :
          ( L
          = ( plus_plus_nat @ K @ N ) ) ) ).

% le_Suc_ex
thf(fact_250_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_251_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_252_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_253_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_254_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_255_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_256_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_257_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_258_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_259_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_260_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_261_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_262_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_263_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_264_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_265_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_266_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_267_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_268_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_269_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_270_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_271_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_272_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_273_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_274_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_275_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_276_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_277_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_278_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_279_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_280_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_281_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_282_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_283_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_284_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_285_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_286_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_287_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_288_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_289_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_290_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_291_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_292_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_293_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_294_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_295_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_296_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ ( suc @ N ) )
                 => ( P @ N ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_297_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ N )
                 => ( P @ ( suc @ N ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_298_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_299_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_300_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_301_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_302_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_303_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_304_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q2: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_305_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_nat @ M5 @ N )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_306_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_307_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_308_discrete,axiom,
    ( ord_less_int
    = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_309_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_310_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_311_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_312_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_313_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_314_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_315_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_316_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_317_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_318_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_319_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_320_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_321_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_322_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_323_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_324_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_325_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_326_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_327_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_328_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_329_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_330_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_331_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_332_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_333_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_334_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_335_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_336_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_337_le__cases3,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_rat @ X @ Z2 )
           => ~ ( ord_less_eq_rat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z2 @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z2 )
               => ~ ( ord_less_eq_rat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z2 @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_338_le__cases3,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z2 ) )
       => ( ( ( ord_less_eq_num @ X @ Z2 )
           => ~ ( ord_less_eq_num @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_num @ Z2 @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z2 )
               => ~ ( ord_less_eq_num @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_num @ Z2 @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_339_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_340_le__cases3,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z2 ) )
       => ( ( ( ord_less_eq_int @ X @ Z2 )
           => ~ ( ord_less_eq_int @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z2 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z2 )
               => ~ ( ord_less_eq_int @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_int @ Z2 @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_341_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y2 )
          & ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_342_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_343_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_344_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_345_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_346_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_347_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_348_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_349_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_350_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_351_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_352_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_353_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_354_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_355_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_356_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_357_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_358_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_359_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_360_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_361_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_362_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_363_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_364_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_365_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_366_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_367_order__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z2 )
       => ( ord_less_eq_rat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_368_order__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_eq_num @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_369_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_370_order__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_371_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: rat,B4: rat] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_372_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: num,B4: num] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_373_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: nat,B4: nat] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_374_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: int,B4: int] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_375_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_376_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_377_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_378_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_379_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_380_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_381_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_382_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_383_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_384_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_385_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_386_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_387_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_388_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_389_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_390_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_391_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_392_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_393_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_394_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_395_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_396_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_397_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_398_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_399_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_400_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_401_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_402_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_403_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_eq_int @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_404_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_405_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_406_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_407_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_eq_int @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_408_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_409_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_410_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_411_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_412_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_413_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_414_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_415_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_416_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_417_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_418_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_419_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_420_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_421_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_422_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_423_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_424_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_425_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_426_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_427_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_428_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_429_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_430_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_431_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_433_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_434_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_435_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_436_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_437_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_438_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_439_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_440_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_441_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_442_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_443_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_444_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_445_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_446_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_447_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_449_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_450_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_451_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_452_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_453_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_454_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_455_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_456_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_457_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_458_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_459_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_460_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_461_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_462_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_463_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_464_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_465_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z: real] :
          ( ( ord_less_real @ X @ Z )
          & ( ord_less_real @ Z @ Y ) ) ) ).

% dense
thf(fact_466_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z: rat] :
          ( ( ord_less_rat @ X @ Z )
          & ( ord_less_rat @ Z @ Y ) ) ) ).

% dense
thf(fact_467_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_468_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_469_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_470_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_471_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_472_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_473_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_474_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_475_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_476_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_477_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_478_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_479_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_480_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_481_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_482_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_483_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_484_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_485_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_486_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_487_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X4 )
             => ( P @ Y4 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_488_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_489_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_490_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_491_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_492_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_493_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_494_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_495_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_496_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_497_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_498_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_499_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_500_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_501_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_502_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_503_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_504_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_505_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_506_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_507_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_508_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N3 )
             => ~ ( P3 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_509_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A4: real,B4: real] :
          ( ( ord_less_real @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: real] : ( P @ A4 @ A4 )
       => ( ! [A4: real,B4: real] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_510_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: rat] : ( P @ A4 @ A4 )
       => ( ! [A4: rat,B4: rat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_511_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: num] : ( P @ A4 @ A4 )
       => ( ! [A4: num,B4: num] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_512_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_513_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B4: int] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_514_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_515_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_516_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_517_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_518_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_519_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_520_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_521_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_522_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_523_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_524_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_525_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_526_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_527_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_528_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_529_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_530_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_531_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_532_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_533_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_543_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_544_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_545_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_546_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_547_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_548_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_549_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_550_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_551_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_552_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_553_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_554_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_555_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_556_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_557_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_558_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_559_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_560_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_561_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_562_order__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_563_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_564_order__less__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_565_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_566_order__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_567_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_568_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_569_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_570_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_571_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_572_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_573_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_574_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_575_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_576_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_577_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_578_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_579_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_580_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_581_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_582_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_583_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_584_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_585_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_586_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_587_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_588_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_589_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_590_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_591_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_592_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_593_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_594_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_595_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_596_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_597_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_598_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_599_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_600_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_601_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_602_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_603_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_604_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_605_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_606_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_607_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_608_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_609_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_610_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_611_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_612_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_613_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_614_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_615_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_616_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_617_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_618_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_619_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_620_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_621_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_622_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_623_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_624_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_625_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_626_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_627_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_628_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_629_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_630_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_631_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_632_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_633_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_634_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_635_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_636_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_637_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_638_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_639_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_640_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_641_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_642_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M7: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M7 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X2: nat] :
                    ( ( P @ X2 )
                   => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_643_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_644_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_645_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_646_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_647_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_648_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_649_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_650_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_651_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_652_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_653_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_654_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_655_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_656_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_657_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_658_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_659_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_660_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_661_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_662_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_663_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_664_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_665_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_666_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_667_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_668_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_669_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_670_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_671_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_672_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ Z2 @ X4 )
         => ( ord_less_eq_real @ Y @ X4 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_673_dense__ge,axiom,
    ! [Z2: rat,Y: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ Z2 @ X4 )
         => ( ord_less_eq_rat @ Y @ X4 ) )
     => ( ord_less_eq_rat @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_674_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_eq_real @ X4 @ Z2 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_le
thf(fact_675_dense__le,axiom,
    ! [Y: rat,Z2: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Y )
         => ( ord_less_eq_rat @ X4 @ Z2 ) )
     => ( ord_less_eq_rat @ Y @ Z2 ) ) ).

% dense_le
thf(fact_676_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_eq_real @ X3 @ Y2 )
          & ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_677_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y2 )
          & ~ ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_678_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ~ ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_679_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_680_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_681_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_682_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_683_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_684_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_685_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_686_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_687_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_real @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_688_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_set_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_689_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_rat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_690_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_num @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_691_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_692_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_int @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_693_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_eq_real @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_694_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_695_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_696_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_697_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_698_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_699_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_700_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_701_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_702_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_703_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_704_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_705_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_706_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_707_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_708_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_709_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_710_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_711_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_eq_real @ A2 @ B2 )
          & ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_712_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_713_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A2 @ B2 )
          & ~ ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_714_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_less_eq_num @ A2 @ B2 )
          & ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_715_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_716_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_717_dense__ge__bounded,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ Z2 @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z2 @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_718_dense__ge__bounded,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z2 @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z2 @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_719_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_720_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_721_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_less_real @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_722_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_set_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_723_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_rat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_724_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_num @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_725_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_726_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_int @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_727_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_less_eq_real @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_728_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_729_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_730_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_731_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_732_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_733_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_734_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_735_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_736_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_737_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_738_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_739_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_740_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_741_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_742_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_743_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_744_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_745_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_less_eq_real @ B2 @ A2 )
          & ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_746_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_747_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A2 )
          & ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_748_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_less_eq_num @ B2 @ A2 )
          & ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_749_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_750_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_751_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_752_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_753_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_754_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_755_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_756_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_757_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_758_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_759_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_760_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_761_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_762_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_763_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_real @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_764_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_set_nat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_765_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_766_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_num @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_767_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_nat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_768_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_int @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_769_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_eq_real @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_770_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_771_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_772_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_773_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_774_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_775_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_776_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_777_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_778_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_779_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_780_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_781_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_782_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_783_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_784_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_785_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_786_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_787_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_788_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_789_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_790_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_791_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_792_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_793_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_794_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_795_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_796_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_797_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_798_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_799_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_800_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_801_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_802_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_803_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_804_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z2 )
       => ( ord_less_set_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_805_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_806_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_807_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_808_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_809_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_real @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_810_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_set_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_811_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_812_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_813_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_814_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_815_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_816_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_817_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_818_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_819_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_820_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_821_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_822_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_823_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_824_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_825_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_826_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_827_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_828_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_829_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_830_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_831_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_832_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_833_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_834_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_835_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_836_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_837_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_838_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_839_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_840_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_841_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_842_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_843_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_844_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_845_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_846_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_847_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_848_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_849_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_850_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_851_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_852_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_853_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_854_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_855_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_856_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_857_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_858_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_859_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_860_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_861_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_862_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_863_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_864_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_865_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_866_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).

% buildup_gives_valid
thf(fact_867_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_868_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_869_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_870_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_871_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_872_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_873_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_874_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_875_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_876_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_877_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_878_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_879_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_880_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ~ ( ord_less_eq_real @ T @ X2 ) ) ).

% minf(8)
thf(fact_881_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).

% minf(8)
thf(fact_882_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z )
     => ~ ( ord_less_eq_num @ T @ X2 ) ) ).

% minf(8)
thf(fact_883_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).

% minf(8)
thf(fact_884_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ~ ( ord_less_eq_int @ T @ X2 ) ) ).

% minf(8)
thf(fact_885_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ord_less_eq_real @ X2 @ T ) ) ).

% minf(6)
thf(fact_886_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ( ord_less_eq_rat @ X2 @ T ) ) ).

% minf(6)
thf(fact_887_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z )
     => ( ord_less_eq_num @ X2 @ T ) ) ).

% minf(6)
thf(fact_888_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ord_less_eq_nat @ X2 @ T ) ) ).

% minf(6)
thf(fact_889_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ord_less_eq_int @ X2 @ T ) ) ).

% minf(6)
thf(fact_890_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ord_less_eq_real @ T @ X2 ) ) ).

% pinf(8)
thf(fact_891_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( ord_less_eq_rat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_892_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z @ X2 )
     => ( ord_less_eq_num @ T @ X2 ) ) ).

% pinf(8)
thf(fact_893_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ord_less_eq_nat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_894_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ord_less_eq_int @ T @ X2 ) ) ).

% pinf(8)
thf(fact_895_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_896_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_897_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_898_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_899_add__0,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ zero_zero_literal @ A )
      = A ) ).

% add_0
thf(fact_900_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_901_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_902_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_903_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_923_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_924_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_925_add_Oright__neutral,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ A @ zero_zero_literal )
      = A ) ).

% add.right_neutral
thf(fact_926_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_927_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_928_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_929_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_930_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_931_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_932_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_933_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_934_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_935_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_936_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_982_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_983_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_984_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_985_zero__reorient,axiom,
    ! [X: literal] :
      ( ( zero_zero_literal = X )
      = ( X = zero_zero_literal ) ) ).

% zero_reorient
thf(fact_986_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_987_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_988_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_989_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_990_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_991_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_992_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_993_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_994_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_995_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_996_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_997_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_998_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_999_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_1000_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1001_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1002_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_1003_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1004_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1005_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1006_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_1007_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_1008_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1009_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_1010_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_1011_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_1012_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_1013_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_1014_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1015_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1016_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1017_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1018_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1019_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_1020_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1021_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1022_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1023_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1024_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X4: nat,Y3: nat] :
              ( ( P @ X4 @ Y3 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_1025_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_1026_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1027_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1028_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1029_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1030_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1031_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ~ ( P @ N )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_1032_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1033_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1034_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_1035_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1036_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_1037_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1038_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1039_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_1040_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_1041_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1042_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1043_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_1044_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1045_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1046_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1047_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1065_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1066_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1067_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1068_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1069_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1070_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1071_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1072_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1073_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1074_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1075_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1076_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1077_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1078_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1079_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1089_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1090_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1091_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1092_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1093_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1094_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1095_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1096_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1109_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1110_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1111_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1112_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_1113_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_1114_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1115_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1116_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1117_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_1118_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1119_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M6: nat] :
            ( N2
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1120_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1121_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1122_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1123_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1124_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1125_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1126_pinf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1127_pinf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1128_pinf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1129_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1130_pinf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1131_pinf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1132_pinf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1133_pinf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1134_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1135_pinf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1136_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1137_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1138_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1139_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1140_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1141_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1142_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1143_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1144_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1145_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1146_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ~ ( ord_less_real @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1147_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ~ ( ord_less_rat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1148_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z @ X2 )
     => ~ ( ord_less_num @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1149_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ~ ( ord_less_nat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1150_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ~ ( ord_less_int @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1151_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ord_less_real @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1152_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( ord_less_rat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1153_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z @ X2 )
     => ( ord_less_num @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1154_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ord_less_nat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1155_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ord_less_int @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1156_minf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1157_minf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1158_minf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1159_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1160_minf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1161_minf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1162_minf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1163_minf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1164_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1165_minf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P4 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P4 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1166_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1167_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1168_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1169_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1170_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1171_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1172_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1173_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1174_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1175_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1176_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ord_less_real @ X2 @ T ) ) ).

% minf(5)
thf(fact_1177_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ( ord_less_rat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1178_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z )
     => ( ord_less_num @ X2 @ T ) ) ).

% minf(5)
thf(fact_1179_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ord_less_nat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1180_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ord_less_int @ X2 @ T ) ) ).

% minf(5)
thf(fact_1181_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ~ ( ord_less_real @ T @ X2 ) ) ).

% minf(7)
thf(fact_1182_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ~ ( ord_less_rat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1183_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z )
     => ~ ( ord_less_num @ T @ X2 ) ) ).

% minf(7)
thf(fact_1184_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ~ ( ord_less_nat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1185_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ~ ( ord_less_int @ T @ X2 ) ) ).

% minf(7)
thf(fact_1186_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1187_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1188_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1189_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1190_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1191_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1192_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1193_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1194_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1195_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1196_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1197_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_1198_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_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_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_1211_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_1212_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_1213_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_1214_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_1215_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_1216_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_1217_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1218_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1219_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ~ ( ord_less_eq_real @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1220_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1221_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z @ X2 )
     => ~ ( ord_less_eq_num @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1222_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1223_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ~ ( ord_less_eq_int @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1224_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_1225_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_1226_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_1227_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_1228_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_1229_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_1230_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( P @ A4 @ B4 )
          = ( P @ B4 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ A4 @ B4 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1231_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N: nat] :
            ( ~ ( P @ N )
            & ( P @ ( suc @ N ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1232_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_1233_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] :
            ( X
           != ( suc @ N ) ) ) ).

% list_decode.cases
thf(fact_1234_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1235_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_1236_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1237_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1238_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_1239_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_1240_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1241_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_1242_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_1243_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_1244_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ Z5 @ X ) )
           => ( ord_less_eq_nat @ Z5 @ Y ) ) ) ) ).

% pred_member
thf(fact_1245_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ X @ Z5 ) )
           => ( ord_less_eq_nat @ Y @ Z5 ) ) ) ) ).

% succ_member
thf(fact_1246_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1247_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1248_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1249_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1250_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1251_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_1252_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_1253_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_1254_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_1255_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1256_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1257_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1258_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1259_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1260_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1261_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1262_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1263_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1264_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
      = ( ord_less_real @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1265_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ( ~ ( ord_less_eq_rat @ B5 @ A5 ) )
      = ( ord_less_rat @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1266_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
      = ( ord_less_num @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1267_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
      = ( ord_less_nat @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1268_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
      = ( ord_less_int @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1269_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1270_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_1271_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_1272_complete__interval,axiom,
    ! [A: real,B: real,P: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: real] :
              ( ( ord_less_eq_real @ A @ C3 )
              & ( ord_less_eq_real @ C3 @ B )
              & ! [X2: real] :
                  ( ( ( ord_less_eq_real @ A @ X2 )
                    & ( ord_less_real @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( ( ord_less_eq_real @ A @ X4 )
                        & ( ord_less_real @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1273_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X2: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X2 )
                    & ( ord_less_nat @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X4 )
                        & ( ord_less_nat @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1274_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: int] :
              ( ( ord_less_eq_int @ A @ C3 )
              & ( ord_less_eq_int @ C3 @ B )
              & ! [X2: int] :
                  ( ( ( ord_less_eq_int @ A @ X2 )
                    & ( ord_less_int @ X2 @ C3 ) )
                 => ( P @ X2 ) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( ( ord_less_eq_int @ A @ X4 )
                        & ( ord_less_int @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1275_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_1276_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_1277_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_1278_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_1279_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info @ ( suc @ ( suc @ N2 ) ) @ TreeList @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_1280_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A2: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ).

% deg1Leaf
thf(fact_1281_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A4: $o,B4: $o] :
          ( T
          = ( vEBT_Leaf @ A4 @ B4 ) ) ) ).

% deg_1_Leaf
thf(fact_1282_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A4: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1283_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1284_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_1285_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_1286_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_1287_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ Xs2 )
              & ( ord_less_nat @ A @ X2 ) ) ) ) ).

% succ_none_empty
thf(fact_1288_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ Xs2 )
              & ( ord_less_nat @ X2 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_1289_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_1290_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% VEBT.inject(2)
thf(fact_1291_obtain__set__succ,axiom,
    ! [X: nat,Z2: nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_nat @ X @ Z2 )
     => ( ( vEBT_VEBT_max_in_set @ A3 @ Z2 )
       => ( ( finite_finite_nat @ B3 )
         => ( ( A3 = B3 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_1292_obtain__set__pred,axiom,
    ! [Z2: nat,X: nat,A3: set_nat] :
      ( ( ord_less_nat @ Z2 @ X )
     => ( ( vEBT_VEBT_min_in_set @ A3 @ Z2 )
       => ( ( finite_finite_nat @ A3 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A3 @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_1293_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_1294_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_1295_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_1296_ex__min__if__finite,axiom,
    ! [S3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ S3 )
            & ~ ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ S3 )
                  & ( ord_le72135733267957522d_enat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1297_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1298_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1299_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1300_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1301_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1302_infinite__growing,axiom,
    ! [X6: set_Extended_enat] :
      ( ( X6 != bot_bo7653980558646680370d_enat )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ X6 )
           => ? [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ X6 )
                & ( ord_le72135733267957522d_enat @ X4 @ Xa ) ) )
       => ~ ( finite4001608067531595151d_enat @ X6 ) ) ) ).

% infinite_growing
thf(fact_1303_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ X6 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X6 )
                & ( ord_less_real @ X4 @ Xa ) ) )
       => ~ ( finite_finite_real @ X6 ) ) ) ).

% infinite_growing
thf(fact_1304_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ X6 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X6 )
                & ( ord_less_rat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X6 ) ) ) ).

% infinite_growing
thf(fact_1305_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( ( member_num @ X4 @ X6 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X6 )
                & ( ord_less_num @ X4 @ Xa ) ) )
       => ~ ( finite_finite_num @ X6 ) ) ) ).

% infinite_growing
thf(fact_1306_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X6 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X6 )
                & ( ord_less_nat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_1307_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ X6 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X6 )
                & ( ord_less_int @ X4 @ Xa ) ) )
       => ~ ( finite_finite_int @ X6 ) ) ) ).

% infinite_growing
thf(fact_1308_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_1309_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_1310_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_1311_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_1312_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1313_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A @ bot_bo7653980558646680370d_enat )
      = ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
thf(fact_1314_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_1315_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_1316_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_1317_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1318_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ bot_bo7653980558646680370d_enat @ A ) ).

% bot.extremum
thf(fact_1319_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_1320_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1321_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1322_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1323_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
      = ( ord_le2529575680413868914d_enat @ bot_bo7653980558646680370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1324_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_1325_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_1326_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_1327_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1328_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] :
      ~ ( ord_le2529575680413868914d_enat @ A @ bot_bo7653980558646680370d_enat ) ).

% bot.extremum_strict
thf(fact_1329_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_1330_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1331_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1332_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1333_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M6: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_nat @ X3 @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1334_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_nat @ X4 @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1335_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M6: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_eq_nat @ X3 @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1336_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_1337_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N2 ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_1338_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_1339_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_1340_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_1341_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_1342_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_1343_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_1344_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_1345_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_1346_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_1347_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_1348_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_1349_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_1350_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_1351_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B4: real] :
      ( ( ord_less_real @ A @ B4 )
      | ( ord_less_real @ B4 @ A ) ) ).

% ex_gt_or_lt
thf(fact_1352_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_1353_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_1354_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_1355_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_1356_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_1357_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1358_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1359_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1360_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1361_List_Ofinite__set,axiom,
    ! [Xs2: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ).

% List.finite_set
thf(fact_1362_List_Ofinite__set,axiom,
    ! [Xs2: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1363_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_1364_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X3: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ X3 @ Y2 )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ X3 @ Z5 )
               => ( ord_less_eq_nat @ Y2 @ Z5 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_1365_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X3: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ Y2 @ X3 )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ Z5 @ X3 )
               => ( ord_less_eq_nat @ Z5 @ Y2 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_1366_finite__has__maximal,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1367_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1368_finite__has__maximal,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1369_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1370_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1371_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1372_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1373_finite__has__minimal,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1374_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1375_finite__has__minimal,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1376_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1377_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1378_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1379_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1380_tvalid,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).

% tvalid
thf(fact_1381_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1382_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1383_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1384_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M6: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M6 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1385_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1386_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_1387_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1388_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_1389_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_1390_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: 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 @ Va2 @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1391_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_1392_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_1393_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_1394_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_1395_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A6: set_nat] :
            ( ( finite_finite_nat @ A6 )
           => ( ! [B6: set_nat] :
                  ( ( ord_less_set_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_1396_finite__psubset__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ! [A6: set_int] :
            ( ( finite_finite_int @ A6 )
           => ( ! [B6: set_int] :
                  ( ( ord_less_set_int @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_1397_finite__psubset__induct,axiom,
    ! [A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [A6: set_complex] :
            ( ( finite3207457112153483333omplex @ A6 )
           => ( ! [B6: set_complex] :
                  ( ( ord_less_set_complex @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_1398_finite__psubset__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [A6: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A6 )
           => ( ! [B6: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_1399_finite__psubset__induct,axiom,
    ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [A6: set_Extended_enat] :
            ( ( finite4001608067531595151d_enat @ A6 )
           => ( ! [B6: set_Extended_enat] :
                  ( ( ord_le2529575680413868914d_enat @ B6 @ A6 )
                 => ( P @ B6 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_1400_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B3 )
      = ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1401_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B3 )
      = ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1402_subset__code_I1_J,axiom,
    ! [Xs2: list_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ B3 )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
           => ( member_set_nat @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1403_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B3 )
      = ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1404_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B3 )
      = ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1405_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B3 )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1406_finite__has__maximal2,axiom,
    ! [A3: set_real,A: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A @ A3 )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A3 )
            & ( ord_less_eq_real @ A @ X4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1407_finite__has__maximal2,axiom,
    ! [A3: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ A @ A3 )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
            & ( ord_le2932123472753598470d_enat @ A @ X4 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1408_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ( ord_less_eq_set_nat @ A @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1409_finite__has__maximal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A3 )
            & ( ord_less_eq_rat @ A @ X4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1410_finite__has__maximal2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A3 )
            & ( ord_less_eq_num @ A @ X4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1411_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( ord_less_eq_nat @ A @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1412_finite__has__maximal2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ( ord_less_eq_int @ A @ X4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1413_finite__has__minimal2,axiom,
    ! [A3: set_real,A: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A @ A3 )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A3 )
            & ( ord_less_eq_real @ X4 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1414_finite__has__minimal2,axiom,
    ! [A3: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ A @ A3 )
       => ? [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
            & ( ord_le2932123472753598470d_enat @ X4 @ A )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1415_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ( ord_less_eq_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1416_finite__has__minimal2,axiom,
    ! [A3: set_rat,A: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A @ A3 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A3 )
            & ( ord_less_eq_rat @ X4 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1417_finite__has__minimal2,axiom,
    ! [A3: set_num,A: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A @ A3 )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A3 )
            & ( ord_less_eq_num @ X4 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1418_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( ord_less_eq_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1419_finite__has__minimal2,axiom,
    ! [A3: set_int,A: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A @ A3 )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A3 )
            & ( ord_less_eq_int @ X4 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1420_finite__list,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_1421_finite__list,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_1422_finite__list,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_1423_finite__list,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_1424_finite__list,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ? [Xs3: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_1425_finite__list,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ? [Xs3: list_Extended_enat] :
          ( ( set_Extended_enat2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_1426_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M5: nat] :
          ( ( ord_less_nat @ K @ M5 )
         => ? [N7: nat] :
              ( ( ord_less_nat @ M5 @ N7 )
              & ( member_nat @ N7 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_1427_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M6: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M6 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1428_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_1429_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_1430_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1431_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic1189837152898106425t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1432_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1433_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1434_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1435_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1436_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic3210252021154270693at_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1437_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1438_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1439_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1440_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1441_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic3210252021154270693at_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1442_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1443_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1444_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1445_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1446_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: extended_enat > num] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic402713867396545063at_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1447_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1448_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1449_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1450_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_1451_count__notin,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ~ ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ( count_list_complex @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1452_count__notin,axiom,
    ! [X: real,Xs2: list_real] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ( count_list_real @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1453_count__notin,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ( count_list_set_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1454_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ( count_list_VEBT_VEBT @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1455_count__notin,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( count_list_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1456_count__notin,axiom,
    ! [X: int,Xs2: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ( count_list_int @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1457_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1458_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X22: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1459_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X22: num] :
      ( ( size_option_num @ X @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1460_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_1461_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_1462_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_1463_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_1464_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_1465_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1466_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1467_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1468_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1469_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1470_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1471_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1472_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1473_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1474_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1475_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1476_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1477_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1478_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1479_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_1480_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_1481_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_1482_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_1483_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1484_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1485_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1486_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1487_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1488_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1489_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1490_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1491_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1492_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1493_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1494_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1495_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1496_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1497_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1498_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1499_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1500_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1501_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1502_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_1503_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_1504_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_1505_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_1506_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_1507_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_1508_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_1509_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1510_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_1511_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_1512_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_1513_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_1514_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1515_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_1516_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_1517_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_1518_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_1519_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_1520_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1521_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1522_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1523_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1524_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_1525_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_1526_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_1527_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_1528_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1529_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1530_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1531_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1532_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1533_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1534_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1535_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1536_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1537_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1538_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1539_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1540_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_1541_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1542_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ M @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_1543_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1544_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_1545_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1546_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1547_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1548_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1549_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% lesseq_shift
thf(fact_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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_1556_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_1557_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_1558_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1559_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1560_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_1561_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_1562_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1563_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1564_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_1565_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_1566_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1567_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_1568_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_1569_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_1570_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1571_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1572_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_1573_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_1574_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_1575_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_1576_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_1577_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_1578_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_1579_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_1580_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_1581_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1582_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1583_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1584_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1585_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_1586_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_1587_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_1588_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1589_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1590_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1591_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1592_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1593_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_1594_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_1595_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1596_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1597_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1598_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1599_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1600_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1601_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1602_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1603_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1604_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1605_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1606_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1607_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1608_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1609_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1610_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1611_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1612_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1613_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1614_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1615_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1616_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1617_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1618_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1619_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1620_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1621_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1622_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_1623_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_1624_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_1625_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).

% mult.commute
thf(fact_1626_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_1627_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_1628_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_1629_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_1630_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1631_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1632_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1633_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1634_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1635_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_1636_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_1637_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X4: real,K2: real] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X4: real,K2: real] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X2: real,K4: real] :
            ( ( ( P @ X2 )
              | ( Q @ X2 ) )
            = ( ( P @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1638_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X4: rat,K2: rat] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X4: rat,K2: rat] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X2: rat,K4: rat] :
            ( ( ( P @ X2 )
              | ( Q @ X2 ) )
            = ( ( P @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1639_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X4: int,K2: int] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X4: int,K2: int] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X2: int,K4: int] :
            ( ( ( P @ X2 )
              | ( Q @ X2 ) )
            = ( ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1640_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X4: real,K2: real] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X4: real,K2: real] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X2: real,K4: real] :
            ( ( ( P @ X2 )
              & ( Q @ X2 ) )
            = ( ( P @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1641_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X4: rat,K2: rat] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X4: rat,K2: rat] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X2: rat,K4: rat] :
            ( ( ( P @ X2 )
              & ( Q @ X2 ) )
            = ( ( P @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1642_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X4: int,K2: int] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X4: int,K2: int] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X2: int,K4: int] :
            ( ( ( P @ X2 )
              & ( Q @ X2 ) )
            = ( ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1643_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_1644_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_1645_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_1646_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1647_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1648_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1649_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1650_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1651_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1652_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_1653_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_1654_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_1655_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1656_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1657_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1658_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( 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 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1659_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( 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 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1660_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( 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 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1661_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_1662_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_1663_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_1664_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_1665_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_1666_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_1667_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_1668_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_1669_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_1670_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_1671_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1672_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1673_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1674_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1675_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1676_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1677_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1678_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1679_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1680_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1681_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1682_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1683_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z3: real] : Y5 = Z3 )
    = ( ^ [A2: real,B2: real] :
          ( ( minus_minus_real @ A2 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1684_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A2: rat,B2: rat] :
          ( ( minus_minus_rat @ A2 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1685_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A2: int,B2: int] :
          ( ( minus_minus_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1686_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1687_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1688_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1689_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1690_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1691_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1692_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1693_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1694_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1695_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1696_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1697_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1698_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1699_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1700_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1701_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1702_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1703_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1704_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1705_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1706_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1707_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1708_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1709_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1710_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1711_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1712_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1713_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1714_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1715_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1716_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1717_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1718_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1719_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1720_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1721_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1722_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1723_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1724_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A3 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1725_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1726_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1727_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1728_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1729_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1730_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_1731_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_1732_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_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_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_1742_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1743_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1744_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1745_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1746_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1747_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1748_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1749_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1750_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_1751_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1752_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_1753_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_1754_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_1755_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_1756_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_1757_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_1758_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_1759_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_1760_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_1761_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_1762_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_1763_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_1764_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1765_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1766_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1767_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1768_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1769_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1770_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1771_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1772_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1773_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1774_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1775_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1776_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1777_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1778_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X @ Z2 ) )
        = ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1779_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X @ Z2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1780_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1781_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z2 ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1782_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1783_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1784_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1785_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1786_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1787_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_1788_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_1789_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_1790_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_1791_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_1792_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1793_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_1794_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1795_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1796_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_1797_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1798_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1799_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_1800_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_1801_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1802_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_1803_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_1804_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1805_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_1806_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_1807_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_1808_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_1809_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_1810_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_1811_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_1812_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_1813_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_1814_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_1815_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_1816_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_1817_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_1818_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_1819_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1820_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1821_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1822_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1823_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1824_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1825_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1826_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_1827_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1828_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1829_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1830_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1831_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1832_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1833_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1834_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_1835_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1836_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1837_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1838_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1839_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1840_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1841_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N2: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
         => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1842_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1843_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1844_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1845_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1846_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1847_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1848_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1849_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1850_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1851_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1852_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1853_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1854_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1855_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_1856_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_1857_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_1858_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_1859_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1860_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1861_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1862_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1863_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_1864_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_1865_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_1866_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_1867_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_1868_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_1869_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_1870_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_1871_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_1872_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_1873_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_1874_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_1875_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_1876_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_1877_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_1878_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_1879_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_1880_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_1881_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_1882_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_1883_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_1884_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_1885_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_1886_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_1887_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_1888_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_1889_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1890_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1891_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1892_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1893_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1894_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1895_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1896_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1897_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1898_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1899_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1900_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_1901_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_1902_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_1903_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1904_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1905_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1906_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_1907_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_1908_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_1909_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_1910_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_1911_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_1912_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1913_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1914_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1915_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1916_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_1917_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_1918_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_1919_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_1920_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1921_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1922_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1923_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1924_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1925_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1926_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1927_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1928_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1929_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1930_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1931_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1932_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1933_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1934_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1935_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1936_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1937_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1938_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1939_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1940_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1941_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1942_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1943_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1944_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1945_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1946_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1947_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1948_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1949_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_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_1960_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_1961_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_1962_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_1963_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_1964_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_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_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_1979_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_1980_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_1981_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_1982_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_1983_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_1984_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_1985_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_1986_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1987_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1988_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1989_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1990_less__1__mult,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N2 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1991_less__1__mult,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1992_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1993_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1994_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1995_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_1996_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_1997_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_1998_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1999_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_2000_set__encode__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( nat_set_encode @ A3 )
            = ( nat_set_encode @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% set_encode_eq
thf(fact_2001_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_2002_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_2003_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_2004_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_2005_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_2006_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_2007_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_2008_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_2009_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_2010_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_2011_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_2012_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_2013_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_2014_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_2015_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2016_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_2017_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2018_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2019_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2020_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2021_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2022_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2023_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2024_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2025_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2026_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2027_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2028_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2029_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2030_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2031_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2032_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2033_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2034_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2035_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2036_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2037_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2038_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2039_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2040_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2041_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2042_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2043_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2044_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2045_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2046_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2047_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2048_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2049_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2050_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2051_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2052_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2053_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2054_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2055_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2056_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2057_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2058_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2059_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2060_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2061_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2062_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2063_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2064_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2065_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2066_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_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_2078_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_2079_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_2080_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2081_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2082_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2083_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2084_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_2085_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_2086_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_2087_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_2088_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D6: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D6 ) )
                & ~ ( P @ D6 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2089_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D6: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D6 ) )
           => ( P @ D6 ) ) ) ) ).

% nat_diff_split
thf(fact_2090_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_2091_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2092_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2093_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_2094_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( nat_set_encode @ A3 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_2095_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2096_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2097_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2098_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2099_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2100_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2101_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2102_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2103_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2104_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2105_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2106_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2107_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2108_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2109_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2110_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2111_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2112_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2113_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2114_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2115_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2116_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2117_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2118_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2119_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z: real] :
          ( ( ord_less_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2120_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z )
         => ( ( ord_less_rat @ Z @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2121_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_2122_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_2123_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_2124_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_2125_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_2126_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_2127_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ 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,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) @ X4 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2128_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_2129_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_2130_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2131_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_2132_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_2133_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_2134_succ__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_2135_pred__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_2136_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% less_shift
thf(fact_2137_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y2: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% greater_shift
thf(fact_2138_succ__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_2139_pred__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_2140_psubsetI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( A3 != B3 )
       => ( ord_less_set_nat @ A3 @ B3 ) ) ) ).

% psubsetI
thf(fact_2141_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2142_psubset__imp__ex__mem,axiom,
    ! [A3: set_complex,B3: set_complex] :
      ( ( ord_less_set_complex @ A3 @ B3 )
     => ? [B4: complex] : ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2143_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B3: set_real] :
      ( ( ord_less_set_real @ A3 @ B3 )
     => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2144_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B3 )
     => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2145_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2146_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2147_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2148_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_2149_psubsetD,axiom,
    ! [A3: set_complex,B3: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A3 @ B3 )
     => ( ( member_complex @ C @ A3 )
       => ( member_complex @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2150_psubsetD,axiom,
    ! [A3: set_real,B3: set_real,C: real] :
      ( ( ord_less_set_real @ A3 @ B3 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2151_psubsetD,axiom,
    ! [A3: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B3 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2152_psubsetD,axiom,
    ! [A3: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2153_psubsetD,axiom,
    ! [A3: set_int,B3: set_int,C: int] :
      ( ( ord_less_set_int @ A3 @ B3 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_2154_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2155_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2156_not__psubset__empty,axiom,
    ! [A3: set_Extended_enat] :
      ~ ( ord_le2529575680413868914d_enat @ A3 @ bot_bo7653980558646680370d_enat ) ).

% not_psubset_empty
thf(fact_2157_not__psubset__empty,axiom,
    ! [A3: set_real] :
      ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_2158_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_2159_not__psubset__empty,axiom,
    ! [A3: set_int] :
      ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_2160_psubsetE,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ~ ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ).

% psubsetE
thf(fact_2161_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% psubset_eq
thf(fact_2162_psubset__imp__subset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_2163_psubset__subset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C4: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C4 )
       => ( ord_less_set_nat @ A3 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_2164_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A7 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_2165_subset__psubset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_set_nat @ B3 @ C4 )
       => ( ord_less_set_nat @ A3 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_2166_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_2167_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2168_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_2169_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_2170_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_2171_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_2172_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_2173_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_2174_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_2175_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_2176_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_2177_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_2178_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_2179_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_2180_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_2181_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_2182_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_2183_mul__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% mul_shift
thf(fact_2184_add__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% add_shift
thf(fact_2185_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_2186_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_2187_mint__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_2188_mint__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_2189_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2190_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2191_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2192_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2193_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2194_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2195_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_2196_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_2197_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_2198_mint__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_2199_maxt__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_2200_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_2201_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_2202_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_2203_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_2204_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_2205_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_2206_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_2207_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_2208_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2209_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_2210_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_2211_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_2212_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_2213_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_2214_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_2215_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_2216_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_2217_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va2: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_2218_mult__less__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2219_mult__less__iff1,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2220_mult__less__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2221_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_2222_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_2223_diff__shunt__var,axiom,
    ! [X: set_Extended_enat,Y: set_Extended_enat] :
      ( ( ( minus_925952699566721837d_enat @ X @ Y )
        = bot_bo7653980558646680370d_enat )
      = ( ord_le7203529160286727270d_enat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2224_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_2225_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_2226_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_2227_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_2228_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_2229_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ zero_zero_nat ) )
     => ( ! [A4: $o,B4: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A4: $o,B4: $o,N: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N ) ) ) )
         => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X4 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X4: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X4 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_2230_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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X4 ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X4 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2231_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A4: $o,Uw2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A4: $o,B4: $o,Va: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_2232_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B4: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_2233_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_2234_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_2235_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_2236_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_2237_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_2238_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_2239_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( A4
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A4
                 => ( ( B4
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B4
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2240_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( B4
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B4
                 => ( ( A4
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A4
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2241_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ 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,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_2242_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
     => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) @ X4 ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X4 ) )
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ X4 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_2243_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( ( B4
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B4
                   => ( ( A4
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A4
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_2244_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( ( A4
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A4
                   => ( ( B4
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B4
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_2245_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2246_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2247_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2248_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2249_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2250_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2251_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_2252_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = A )
      = ( ( member_real @ ( minus_minus_real @ X @ A ) @ ring_1_Ints_real )
        & ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ord_less_real @ A @ one_one_real ) ) ) ).

% frac_unique_iff
thf(fact_2253_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ord_less_rat @ A @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_2254_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_2255_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_2256_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_2257_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_2258_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_2259_frac__in__Ints__iff,axiom,
    ! [X: real] :
      ( ( member_real @ ( archim2898591450579166408c_real @ X ) @ ring_1_Ints_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% frac_in_Ints_iff
thf(fact_2260_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_2261_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_2262_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_2263_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_2264_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% frac_eq_0_iff
thf(fact_2265_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = zero_zero_rat )
      = ( member_rat @ X @ ring_1_Ints_rat ) ) ).

% frac_eq_0_iff
thf(fact_2266_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) )
      = ( ~ ( member_real @ X @ ring_1_Ints_real ) ) ) ).

% frac_gt_0_iff
thf(fact_2267_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) )
      = ( ~ ( member_rat @ X @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_2268_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_2269_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_2270_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_2271_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_2272_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_2273_Ints__0,axiom,
    member_complex @ zero_zero_complex @ ring_1_Ints_complex ).

% Ints_0
thf(fact_2274_Ints__0,axiom,
    member_real @ zero_zero_real @ ring_1_Ints_real ).

% Ints_0
thf(fact_2275_Ints__0,axiom,
    member_rat @ zero_zero_rat @ ring_1_Ints_rat ).

% Ints_0
thf(fact_2276_Ints__0,axiom,
    member_int @ zero_zero_int @ ring_1_Ints_int ).

% Ints_0
thf(fact_2277_Ints__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B @ ring_1_Ints_complex )
       => ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).

% Ints_mult
thf(fact_2278_Ints__mult,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B @ ring_1_Ints_real )
       => ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).

% Ints_mult
thf(fact_2279_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( times_times_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_mult
thf(fact_2280_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_mult
thf(fact_2281_Ints__add,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B @ ring_1_Ints_complex )
       => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).

% Ints_add
thf(fact_2282_Ints__add,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B @ ring_1_Ints_real )
       => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).

% Ints_add
thf(fact_2283_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_add
thf(fact_2284_Ints__add,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_add
thf(fact_2285_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_2286_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_2287_Ints__1,axiom,
    member_real @ one_one_real @ ring_1_Ints_real ).

% Ints_1
thf(fact_2288_Ints__1,axiom,
    member_complex @ one_one_complex @ ring_1_Ints_complex ).

% Ints_1
thf(fact_2289_Ints__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B @ ring_1_Ints_complex )
       => ( member_complex @ ( minus_minus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).

% Ints_diff
thf(fact_2290_Ints__diff,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B @ ring_1_Ints_real )
       => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).

% Ints_diff
thf(fact_2291_Ints__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( minus_minus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_diff
thf(fact_2292_Ints__diff,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_diff
thf(fact_2293_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( ( plus_plus_complex @ A @ A )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2294_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ( plus_plus_real @ A @ A )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2295_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A @ A )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2296_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A @ A )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_2297_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
       != zero_zero_complex ) ) ).

% Ints_odd_nonzero
thf(fact_2298_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
       != zero_zero_real ) ) ).

% Ints_odd_nonzero
thf(fact_2299_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_2300_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_2301_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_2302_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_2303_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
        = ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% Ints_odd_less_0
thf(fact_2304_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
        = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_2305_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_2306_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( 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 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_2307_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_2308_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( 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 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_2309_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_2310_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2311_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2312_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2313_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2314_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_2315_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z2 ) ) ).

% pair_less_iff1
thf(fact_2316_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
        = A3 ) ) ).

% set_encode_inverse
thf(fact_2317_frac__neg,axiom,
    ! [X: real] :
      ( ( ( member_real @ X @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
          = zero_zero_real ) )
      & ( ~ ( member_real @ X @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X ) ) ) ) ) ).

% frac_neg
thf(fact_2318_frac__neg,axiom,
    ! [X: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).

% frac_neg
thf(fact_2319_power__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% power_shift
thf(fact_2320_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_2321_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_2322_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_2323_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_2324_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_2325_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_2326_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_2327_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_2328_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_2329_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_2330_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_2331_verit__minus__simplify_I4_J,axiom,
    ! [B: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_2332_verit__minus__simplify_I4_J,axiom,
    ! [B: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_2333_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_2334_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_2335_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_2336_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_2337_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_2338_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_2339_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_2340_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_2341_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_2342_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_2343_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_2344_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_2345_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_2346_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_2347_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_2348_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_2349_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_2350_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_2351_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_2352_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_2353_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_2354_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_2355_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_2356_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_2357_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_2358_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_2359_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_2360_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_2361_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_2362_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_2363_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_2364_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_2365_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_2366_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_2367_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_2368_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_2369_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_2370_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_2371_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_2372_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_2373_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_2374_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_2375_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_2376_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_2377_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_2378_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_2379_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_2380_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_2381_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_2382_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_2383_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_2384_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_2385_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_2386_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_2387_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_2388_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_2389_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_2390_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_2391_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_2392_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_2393_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_2394_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_2395_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_2396_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_2397_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_2398_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_2399_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_2400_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ one_one_real @ N2 )
      = one_one_real ) ).

% power_one
thf(fact_2401_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_2402_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ one_one_complex @ N2 )
      = one_one_complex ) ).

% power_one
thf(fact_2403_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_2404_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_2405_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_2406_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_2407_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_2408_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2409_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2410_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_2411_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2412_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2413_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2414_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2415_set__decode__inverse,axiom,
    ! [N2: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N2 ) )
      = N2 ) ).

% set_decode_inverse
thf(fact_2416_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_2417_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_2418_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_2419_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_2420_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_2421_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_2422_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_2423_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_2424_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_2425_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_2426_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_2427_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_2428_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_2429_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_2430_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_2431_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_2432_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_2433_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_2434_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_2435_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_2436_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_2437_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_2438_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_2439_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_2440_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_2441_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_2442_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_2443_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_2444_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_2445_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_2446_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_2447_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_2448_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_2449_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_2450_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_2451_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_2452_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_2453_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_2454_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_2455_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_2456_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_2457_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_2458_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_2459_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_2460_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_2461_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_2462_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_2463_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_2464_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_2465_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_2466_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_2467_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_2468_power__inject__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2469_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2470_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2471_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2472_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_2473_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2474_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2475_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2476_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2477_mult__minus1,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
      = ( uminus_uminus_int @ Z2 ) ) ).

% mult_minus1
thf(fact_2478_mult__minus1,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
      = ( uminus_uminus_real @ Z2 ) ) ).

% mult_minus1
thf(fact_2479_mult__minus1,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z2 )
      = ( uminus1482373934393186551omplex @ Z2 ) ) ).

% mult_minus1
thf(fact_2480_mult__minus1,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
      = ( uminus_uminus_rat @ Z2 ) ) ).

% mult_minus1
thf(fact_2481_mult__minus1,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
      = ( uminus1351360451143612070nteger @ Z2 ) ) ).

% mult_minus1
thf(fact_2482_mult__minus1__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z2 ) ) ).

% mult_minus1_right
thf(fact_2483_mult__minus1__right,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z2 ) ) ).

% mult_minus1_right
thf(fact_2484_mult__minus1__right,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z2 ) ) ).

% mult_minus1_right
thf(fact_2485_mult__minus1__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z2 ) ) ).

% mult_minus1_right
thf(fact_2486_mult__minus1__right,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z2 ) ) ).

% mult_minus1_right
thf(fact_2487_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_2488_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_2489_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_2490_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_2491_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_2492_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_2493_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_2494_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_2495_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_2496_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_2497_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2498_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2499_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2500_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2501_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_2502_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_2503_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_2504_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_2505_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_2506_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_2507_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_2508_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_2509_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_2510_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_2511_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_2512_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_2513_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_2514_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_2515_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_2516_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_2517_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_2518_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_2519_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_2520_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_2521_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_2522_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_2523_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_2524_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_2525_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_2526_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_2527_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2528_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2529_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2530_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2531_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2532_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_2533_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_2534_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_2535_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_2536_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( power_power_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_2537_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_2538_power__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( power_power_real @ A @ N2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_2539_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ( power_power_int @ A @ N2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_2540_power__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( power_power_complex @ A @ N2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_2541_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_2542_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_2543_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_2544_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_2545_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_2546_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2547_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2548_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2549_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2550_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_2551_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_2552_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_2553_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_2554_power__mono__iff,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2555_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2556_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2557_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2558_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_2559_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_2560_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_2561_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_2562_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_2563_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_2564_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_2565_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_2566_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_2567_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_2568_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2569_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2570_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2571_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2572_power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2573_power__minus,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2574_power__minus,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2575_power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2576_power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2577_power__not__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N2 )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_2578_power__not__zero,axiom,
    ! [A: nat,N2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N2 )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_2579_power__not__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N2 )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_2580_power__not__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N2 )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_2581_power__not__zero,axiom,
    ! [A: complex,N2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N2 )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_2582_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_2583_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_2584_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_2585_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_2586_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_2587_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_2588_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_2589_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_2590_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_2591_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_2592_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_2593_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_2594_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_2595_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_2596_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_2597_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_2598_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_2599_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_2600_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_2601_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_2602_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_2603_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_2604_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_2605_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_2606_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_2607_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_2608_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_2609_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_2610_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_2611_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_2612_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_2613_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_2614_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_2615_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_2616_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_2617_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_2618_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_2619_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X2 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_2620_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N: nat] :
        ! [X2: list_o] :
          ( ( member_list_o @ X2 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X2 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_2621_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N: nat] :
        ! [X2: list_nat] :
          ( ( member_list_nat @ X2 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_2622_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2623_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2624_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_2625_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2626_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2627_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A: complex] :
      ( ( A3
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2628_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2629_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2630_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_2631_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_2632_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_2633_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_2634_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_2635_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_2636_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_2637_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_2638_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_2639_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_2640_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_2641_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_2642_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_2643_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_2644_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_2645_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_2646_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_2647_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_2648_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_2649_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_2650_minus__diff__minus,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_2651_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_2652_minus__diff__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_2653_minus__diff__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_2654_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_2655_Ints__power,axiom,
    ! [A: real,N2: nat] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( member_real @ ( power_power_real @ A @ N2 ) @ ring_1_Ints_real ) ) ).

% Ints_power
thf(fact_2656_Ints__power,axiom,
    ! [A: int,N2: nat] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( power_power_int @ A @ N2 ) @ ring_1_Ints_int ) ) ).

% Ints_power
thf(fact_2657_Ints__power,axiom,
    ! [A: complex,N2: nat] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( member_complex @ ( power_power_complex @ A @ N2 ) @ ring_1_Ints_complex ) ) ).

% Ints_power
thf(fact_2658_minus__in__Ints__iff,axiom,
    ! [X: int] :
      ( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
      = ( member_int @ X @ ring_1_Ints_int ) ) ).

% minus_in_Ints_iff
thf(fact_2659_minus__in__Ints__iff,axiom,
    ! [X: real] :
      ( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% minus_in_Ints_iff
thf(fact_2660_minus__in__Ints__iff,axiom,
    ! [X: complex] :
      ( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
      = ( member_complex @ X @ ring_1_Ints_complex ) ) ).

% minus_in_Ints_iff
thf(fact_2661_minus__in__Ints__iff,axiom,
    ! [X: rat] :
      ( ( member_rat @ ( uminus_uminus_rat @ X ) @ ring_1_Ints_rat )
      = ( member_rat @ X @ ring_1_Ints_rat ) ) ).

% minus_in_Ints_iff
thf(fact_2662_minus__in__Ints__iff,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ring_11222124179247155820nteger )
      = ( member_Code_integer @ X @ ring_11222124179247155820nteger ) ) ).

% minus_in_Ints_iff
thf(fact_2663_Ints__minus,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).

% Ints_minus
thf(fact_2664_Ints__minus,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).

% Ints_minus
thf(fact_2665_Ints__minus,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).

% Ints_minus
thf(fact_2666_Ints__minus,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( member_rat @ ( uminus_uminus_rat @ A ) @ ring_1_Ints_rat ) ) ).

% Ints_minus
thf(fact_2667_Ints__minus,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( member_Code_integer @ ( uminus1351360451143612070nteger @ A ) @ ring_11222124179247155820nteger ) ) ).

% Ints_minus
thf(fact_2668_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_2669_power__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2670_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2671_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2672_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2673_zero__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2674_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2675_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2676_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2677_zero__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2678_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2679_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2680_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2681_finite__set__decode,axiom,
    ! [N2: nat] : ( finite_finite_nat @ ( nat_set_decode @ N2 ) ) ).

% finite_set_decode
thf(fact_2682_one__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2683_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2684_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2685_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2686_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_2687_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_2688_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_2689_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_2690_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_2691_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_2692_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_2693_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_2694_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_2695_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_2696_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_2697_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_2698_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_2699_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_2700_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_2701_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_2702_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_2703_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_2704_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_2705_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_2706_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_2707_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_2708_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_2709_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_2710_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_2711_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_2712_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_2713_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_2714_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_2715_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_2716_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_2717_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_2718_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_2719_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_2720_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_2721_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_2722_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_2723_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_2724_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_2725_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_2726_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_2727_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_2728_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_2729_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_2730_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_2731_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_2732_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_2733_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_2734_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_2735_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_2736_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_2737_power__Suc2,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2738_power__Suc2,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2739_power__Suc2,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2740_power__Suc2,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2741_power__Suc2,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2742_power__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2743_power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2744_power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2745_power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2746_power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2747_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_2748_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_2749_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_2750_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_2751_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_2752_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2753_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2754_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2755_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2756_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2757_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2758_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2759_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2760_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2761_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2762_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_2763_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_2764_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_2765_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_2766_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_2767_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_2768_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_2769_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_2770_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_2771_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_2772_power__add,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_add
thf(fact_2773_power__add,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_add
thf(fact_2774_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2775_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2776_power__add,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_2777_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_2778_count__le__length,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs2 @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% count_le_length
thf(fact_2779_count__le__length,axiom,
    ! [Xs2: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs2 @ X ) @ ( size_size_list_o @ Xs2 ) ) ).

% count_le_length
thf(fact_2780_count__le__length,axiom,
    ! [Xs2: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X ) @ ( size_size_list_nat @ Xs2 ) ) ).

% count_le_length
thf(fact_2781_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_2782_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2783_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2784_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2785_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2786_power__le__one,axiom,
    ! [A: real,N2: 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 @ N2 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2787_power__le__one,axiom,
    ! [A: rat,N2: 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 @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2788_power__le__one,axiom,
    ! [A: nat,N2: 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 @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2789_power__le__one,axiom,
    ! [A: int,N2: 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 @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2790_power__inject__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N2 ) )
        = ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2791_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
        = ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2792_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2793_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2794_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2795_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2796_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2797_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2798_power__less__power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2799_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2800_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2801_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2802_power__gt1__lemma,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2803_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2804_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2805_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2806_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_2807_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_2808_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_2809_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_2810_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_2811_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_2812_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_2813_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_2814_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_2815_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_2816_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_2817_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_2818_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_2819_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_2820_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_2821_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_2822_power__gt1,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2823_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2824_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2825_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2826_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2827_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2828_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_2829_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2830_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_2831_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2832_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2833_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2834_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2835_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2836_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2837_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2838_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2839_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2840_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2841_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2842_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2843_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_rat @ zero_zero_rat @ N2 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_2844_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_nat @ zero_zero_nat @ N2 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_2845_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ zero_zero_real @ N2 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_2846_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_int @ zero_zero_int @ N2 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_2847_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ zero_zero_complex @ N2 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_2848_length__pos__if__in__set,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2849_length__pos__if__in__set,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2850_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2851_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2852_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2853_length__pos__if__in__set,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2854_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2855_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_2856_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_2857_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_2858_power__Suc__less,axiom,
    ! [A: real,N2: 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 @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2859_power__Suc__less,axiom,
    ! [A: rat,N2: 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 @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2860_power__Suc__less,axiom,
    ! [A: nat,N2: 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 @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2861_power__Suc__less,axiom,
    ! [A: int,N2: 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 @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2862_power__Suc__le__self,axiom,
    ! [A: real,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2863_power__Suc__le__self,axiom,
    ! [A: rat,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2864_power__Suc__le__self,axiom,
    ! [A: nat,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2865_power__Suc__le__self,axiom,
    ! [A: int,N2: 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 @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2866_power__Suc__less__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2867_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2868_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2869_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2870_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2871_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2872_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2873_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2874_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2875_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2876_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2877_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ 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 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2878_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2879_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2880_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2881_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2882_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N2 )
              = ( power_power_real @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2883_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N2 )
              = ( power_power_rat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2884_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2885_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2886_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ N2 )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2887_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N2 )
        = ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2888_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2889_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2890_self__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2891_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2892_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2893_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2894_one__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2895_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2896_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2897_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2898_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_2899_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_2900_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_2901_power__strict__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2902_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2903_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2904_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2905_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2906_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2907_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2908_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2909_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2910_power__minus__mult,axiom,
    ! [N2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2911_power__minus__mult,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2912_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2913_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2914_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2915_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2916_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2917_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2918_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2919_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2920_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_2921_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_2922_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2923_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X4: real] :
            ( ( ord_less_real @ zero_zero_real @ X4 )
            & ( ( power_power_real @ X4 @ N2 )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N2 )
                    = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2924__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X2: nat] :
              ( ( ( ( vEBT_VEBT_high @ X2 @ na )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X2 )
                & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).

% "5.hyps"(9)
thf(fact_2925_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2926_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2927_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2928_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2929_power__minus_H,axiom,
    ! [X: int,N2: nat] :
      ( ( nO_MATCH_int_int @ one_one_int @ X )
     => ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N2 )
        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2930_power__minus_H,axiom,
    ! [X: real,N2: nat] :
      ( ( nO_MATCH_real_real @ one_one_real @ X )
     => ( ( power_power_real @ ( uminus_uminus_real @ X ) @ N2 )
        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2931_power__minus_H,axiom,
    ! [X: complex,N2: nat] :
      ( ( nO_MAT8947977539597988553omplex @ one_one_complex @ X )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N2 )
        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2932_power__minus_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( nO_MATCH_rat_rat @ one_one_rat @ X )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N2 )
        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2933_power__minus_H,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N2 )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2934_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
      = ( nat_prod_decode_aux @ K @ M ) ) ).

% prod_decode_triangle_add
thf(fact_2935_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N2: nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2936_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N2: nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2937_inthall,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,N2: nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( P @ ( nth_set_nat @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2938_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N2: nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2939_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2940_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N2: nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2941_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_2942__C5_Ohyps_C_I2_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "5.hyps"(2)
thf(fact_2943_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numera1916890842035813515d_enat @ M )
        = ( numera1916890842035813515d_enat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_2944_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_2945_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_2946_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_2947_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_2948_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(78)
thf(fact_2949_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_2950_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_2951_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_2952_negative__eq__positive,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_2953__C5_Ohyps_C_I8_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.hyps"(8)
thf(fact_2954_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_2955_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_2956_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_2957_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_2958_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_2959_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_2960_prod__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_prod_decode @ X )
        = ( nat_prod_decode @ Y ) )
      = ( X = Y ) ) ).

% prod_decode_eq
thf(fact_2961_dp,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% dp
thf(fact_2962_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_2963__C5_Ohyps_C_I5_J,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X7 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).

% "5.hyps"(5)
thf(fact_2964_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_2965_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_2966_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_2967_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_2968_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_2969_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_2970_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_2971_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_2972_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_2973_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_2974_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_2975_high__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = Y ) ) ).

% high_inv
thf(fact_2976_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2977_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2978_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2979_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2980_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2981_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2982_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2983_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z2 ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2984_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2985_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2986_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2987_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2988_low__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = X ) ) ).

% low_inv
thf(fact_2989_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_2990_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_2991_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_2992_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_2993_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_2994_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_2995_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2996_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2997_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2998_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2999_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_3000_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_3001_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_3002_num__double,axiom,
    ! [N2: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_3003_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_3004_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_3005_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_3006_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_3007_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_3008_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D6: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D6 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_3009_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_3010_misiz,axiom,
    ! [T: vEBT_VEBT,N2: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% misiz
thf(fact_3011_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_3012_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: 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 @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_3013_helpyd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpyd
thf(fact_3014_helpypredd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpypredd
thf(fact_3015_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_3016_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_3017_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_3018_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_3019__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
    ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% \<open>ma \<le> 2 ^ deg\<close>
thf(fact_3020_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_3021_xdegrel,axiom,
    ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% xdegrel
thf(fact_3022_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( 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_3023_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_3024_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_3025_prod__decode__inverse,axiom,
    ! [N2: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode @ N2 ) )
      = N2 ) ).

% prod_decode_inverse
thf(fact_3026_prod__encode__inverse,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
      = X ) ).

% prod_encode_inverse
thf(fact_3027_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3028_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3029_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3030_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_3031_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_3032_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_3033_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_3034_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_3035_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_3036_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_3037_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3038_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3039_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3040_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3041_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3042_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3043_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3044_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3045_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3046_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_3047_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_3048_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_on7984719198319812577d_enat
        = ( numera1916890842035813515d_enat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_3049_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_3050_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_3051_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_3052_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_3053_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_3054_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera1916890842035813515d_enat @ N2 )
        = one_on7984719198319812577d_enat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_3055_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera6690914467698888265omplex @ N2 )
        = one_one_complex )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_3056_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_real @ N2 )
        = one_one_real )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_3057_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_3058_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_3059_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3060_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3061_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3062_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3063_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3064_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3065_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3066_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3067_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3068_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3069_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3070_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3071_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3072_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3073_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3074_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3075_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3076_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3077_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3078_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3079_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3080_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3081_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3082_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_3083_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_3084_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_3085_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_3086_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_3087_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3088_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3089_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3090_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3091_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3092_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3093_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3094_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3095_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3096_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3097_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3098_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3099_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3100_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3101_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3102_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_3103_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_3104_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_3105_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_3106_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_3107_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_3108_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_3109_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_3110_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_3111_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_3112_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_3113_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3114_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3115_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3116_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3117_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3118_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3119_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3120_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3121_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3122_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3123_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3124_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3125_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3126_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3127_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N2 ) )
      = ( numera1916890842035813515d_enat @ N2 ) ) ).

% of_nat_numeral
thf(fact_3128_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6690914467698888265omplex @ N2 ) ) ).

% of_nat_numeral
thf(fact_3129_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% of_nat_numeral
thf(fact_3130_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% of_nat_numeral
thf(fact_3131_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% of_nat_numeral
thf(fact_3132_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% of_nat_numeral
thf(fact_3133_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3134_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3135_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3136_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3137_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_add
thf(fact_3138_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_add
thf(fact_3139_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_add
thf(fact_3140_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_3141_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_3142_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3143_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3144_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3145_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3146_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3147_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3148_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ N2 )
        = one_one_real )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3149_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ N2 )
        = one_one_rat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3150_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3151_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3152_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3153_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3154_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3155_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3156_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3157_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_3158_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_3159_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_3160_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3161_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3162_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_3163_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_3164_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_3165_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_3166_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_3167_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_3168_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_3169_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_3170_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_3171_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_3172_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_3173_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_3174_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_3175_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_3176_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_3177_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_3178_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_3179_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_3180_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_3181_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_3182_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_3183_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_3184_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3185_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3186_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3187_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3188_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3189_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3190_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3191_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3192_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3193_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3194_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3195_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3196_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3197_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3198_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_3199_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_3200_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3201_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3202_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3203_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3204_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3205_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3206_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3207_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3208_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3209_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3210_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3211_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3212_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3213_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3214_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3215_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3216_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3217_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_3218_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_3219_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_3220_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_3221_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_3222_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3223_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3224_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3225_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3226_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3227_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_3228_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_3229_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_3230_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_3231_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_3232_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3233_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3234_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3235_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3236_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3237_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3238_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3239_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3240_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3241_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3242_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3243_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3244_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3245_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3246_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_3247_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_3248_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_3249_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_3250_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_3251_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_3252_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_3253_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_3254_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_3255_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_3256_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_3257_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_3258_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_3259_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_3260_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_3261_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_3262_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_3263_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_3264_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_3265_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_3266_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_3267_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_3268_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_3269_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_3270_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_3271_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_3272_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_3273_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_3274_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_3275_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_3276_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_3277_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_3278_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_3279_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3280_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3281_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3282_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3283_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3284_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3285_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3286_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3287_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3288_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3289_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_3290_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_3291_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_3292_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_3293_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_3294_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_3295_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_3296_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_3297_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_3298_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3299_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3300_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3301_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3302_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3303_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3304_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3305_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3306_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3307_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3308_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3309_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3310_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3311_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3312_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3313_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3314_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N: nat] : ( ord_less_real @ ( power_power_real @ X @ N ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3315_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_3316_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_3317_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_3318_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_3319_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_3320_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3321_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3322_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3323_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N ) ) ) ).

% real_arch_pow
thf(fact_3324_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_3325_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_3326_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_3327_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_3328_Bernoulli__inequality,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).

% Bernoulli_inequality
thf(fact_3329_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_3330_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y4: real] :
        ? [N: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_3331_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_3332_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_3333_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_3334_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_3335_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_3336_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_3337_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_3338_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_3339_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3340_mult__numeral__1__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3341_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3342_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3343_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3344_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3345_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3346_mult__numeral__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3347_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3348_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3349_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3350_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3351_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_3352_numeral__One,axiom,
    ( ( numera1916890842035813515d_enat @ one )
    = one_on7984719198319812577d_enat ) ).

% numeral_One
thf(fact_3353_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_3354_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_3355_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_3356_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_3357_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_3358_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_3359_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_3360_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3361_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_real @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_3362_mult__2,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3363_mult__2,axiom,
    ! [Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3364_mult__2,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3365_mult__2,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3366_mult__2,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3367_mult__2,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3368_mult__2__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3369_mult__2__right,axiom,
    ! [Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3370_mult__2__right,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3371_mult__2__right,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3372_mult__2__right,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3373_mult__2__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3374_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_3375_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_3376_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_3377_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_3378_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_3379_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_3380_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_3381_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_3382_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_3383_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_3384_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_3385_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_3386_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_3387_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_3388_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_3389_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_3390_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_3391_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_3392_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_3393_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_3394_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_3395_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_3396_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_3397_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_3398_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_3399_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_3400_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_3401_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_3402_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_3403_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_3404_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_3405_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_3406_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_3407_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_3408_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_3409_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_3410_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_3411_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_3412_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_3413_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_3414_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_3415_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_3416_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_3417_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_3418_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_3419_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_3420_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3421_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3422_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3423_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3424_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3425_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_3426_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_3427_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3428_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3429_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3430_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3431_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3432_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3433_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3434_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3435_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3436_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3437_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_3438_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_3439_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_3440_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_3441_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_3442_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_3443_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_3444_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : Y5 = Z3 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
                = ( nth_VEBT_VEBT @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3445_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z3: list_o] : Y5 = Z3 )
    = ( ^ [Xs: list_o,Ys2: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I3 )
                = ( nth_o @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3446_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z3: list_nat] : Y5 = Z3 )
    = ( ^ [Xs: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I3 )
                = ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3447_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X7: vEBT_VEBT] : ( P @ I3 @ X7 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3448_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X7: $o] : ( P @ I3 @ X7 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3449_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X7: nat] : ( P @ I3 @ X7 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3450_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) )
       => ( Xs2 = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_3451_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys3: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys3 @ I2 ) ) )
       => ( Xs2 = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_3452_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys3: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys3 @ I2 ) ) )
       => ( Xs2 = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_3453_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% real_arch_simple
thf(fact_3454_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% real_arch_simple
thf(fact_3455_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% reals_Archimedean2
thf(fact_3456_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% reals_Archimedean2
thf(fact_3457_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_3458_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_3459_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_3460_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_3461_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_3462_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z5: int] :
        ? [N3: nat] :
          ( Z5
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3463_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_3464_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_3465_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_3466_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_3467_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% negD
thf(fact_3468_int__of__nat__induct,axiom,
    ! [P: int > $o,Z2: int] :
      ( ! [N: nat] : ( P @ ( semiri1314217659103216013at_int @ N ) )
     => ( ! [N: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) )
       => ( P @ Z2 ) ) ) ).

% int_of_nat_induct
thf(fact_3469_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% int_cases
thf(fact_3470_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_3471_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_3472_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_3473_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_3474_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_3475_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_3476_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).

% nat_less_real_le
thf(fact_3477_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_3478_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_3479_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_3480_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N3: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3481_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_3482_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_3483_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_3484_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_3485_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_3486_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_3487_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_3488_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_3489_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_3490_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_3491_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3492_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3493_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3494_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3495_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3496_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_3497_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_3498_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_3499_Ints__of__nat,axiom,
    ! [N2: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ring_1_Ints_complex ) ).

% Ints_of_nat
thf(fact_3500_Ints__of__nat,axiom,
    ! [N2: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N2 ) @ ring_1_Ints_real ) ).

% Ints_of_nat
thf(fact_3501_Ints__of__nat,axiom,
    ! [N2: nat] : ( member_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ring_1_Ints_rat ) ).

% Ints_of_nat
thf(fact_3502_Ints__of__nat,axiom,
    ! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ring_1_Ints_int ) ).

% Ints_of_nat
thf(fact_3503_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_3504_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_3505_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_3506_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_3507_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_3508_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_3509_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_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_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_3517_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_3518_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_3519_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_3520_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_3521_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_3522_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_3523_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_3524_zero__le__even__power_H,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_3525_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_3526_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_3527_power__odd__eq,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3528_power__odd__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3529_power__odd__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3530_power__odd__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3531_power__odd__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3532_Ints__numeral,axiom,
    ! [N2: num] : ( member_complex @ ( numera6690914467698888265omplex @ N2 ) @ ring_1_Ints_complex ) ).

% Ints_numeral
thf(fact_3533_Ints__numeral,axiom,
    ! [N2: num] : ( member_real @ ( numeral_numeral_real @ N2 ) @ ring_1_Ints_real ) ).

% Ints_numeral
thf(fact_3534_Ints__numeral,axiom,
    ! [N2: num] : ( member_int @ ( numeral_numeral_int @ N2 ) @ ring_1_Ints_int ) ).

% Ints_numeral
thf(fact_3535_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3536_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3537_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_3538_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_3539_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3540_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3541_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3542_odd__power__less__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_3543_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_3544_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_3545_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_3546_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_3547_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_3548_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_3549_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_3550_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3551_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3552_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3553_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3554_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_3555_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_3556_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_3557_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_3558_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3559_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3560_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3561_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3562_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I2 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3563_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3564_all__nth__imp__all__set,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,X: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
         => ( P @ ( nth_set_nat @ Xs2 @ I2 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3565_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3566_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3567_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3568_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3569_in__set__conv__nth,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3570_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3571_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3572_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3573_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3574_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3575_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3576_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_int @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3577_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3578_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_o @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3579_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3580_nth__mem,axiom,
    ! [N2: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N2 ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3581_nth__mem,axiom,
    ! [N2: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N2 ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3582_nth__mem,axiom,
    ! [N2: nat,Xs2: list_set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs2 @ N2 ) @ ( set_set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3583_nth__mem,axiom,
    ! [N2: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N2 ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3584_nth__mem,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3585_nth__mem,axiom,
    ! [N2: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N2 ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3586_nth__mem,axiom,
    ! [N2: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N2 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3587_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_3588_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_3589_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_3590_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_3591_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_3592_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3593_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3594_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3595_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3596_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3597_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3598_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3599_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3600_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_3601_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_3602_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_3603_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_3604_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_3605_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_3606_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_3607_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_3608_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_3609_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_le_numeral
thf(fact_3610_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_3611_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_3612_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_3613_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_3614_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_3615_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_3616_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_3617_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_3618_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_3619_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_3620_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_less_numeral
thf(fact_3621_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_3622_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_3623_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_3624_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M5: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M5 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_3625_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% one_le_numeral
thf(fact_3626_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_3627_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_3628_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_3629_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_3630_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_3631_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N )
          & ( K
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_3632_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% pos_int_cases
thf(fact_3633_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% int_cases4
thf(fact_3634_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) )
       => ~ ! [N: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% int_cases3
thf(fact_3635_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% neg_int_cases
thf(fact_3636_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_3637_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_3638_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_3639_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_3640_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_3641_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3642_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3643_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3644_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3645_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_3646_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_3647_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_3648_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_3649_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3650_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3651_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3652_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3653_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3654_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3655_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3656_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3657_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3658_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_3659_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_3660_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_3661_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_3662_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_3663_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_3664_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_3665_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_3666_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_3667_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_3668_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ N2 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_3669_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ N2 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_3670_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ N2 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_3671_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ N2 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_3672_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ N2 )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_3673_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_3674_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_3675_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_3676_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_3677_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_3678_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_3679_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_3680_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
                                   => ( ( 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 @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_3681_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3682_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3683_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3684_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3685_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3686_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3687_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3688_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3689_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3690_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3691_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3692_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_3693_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_3694_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_3695_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_3696_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3697_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3698_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3699_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3700_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_3701_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_3702_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_3703_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_3704_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_3705_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_3706_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_3707_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_3708_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_3709_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_3710_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_3711_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_3712_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_3713_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_3714_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_3715_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_3716_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_3717_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_3718_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_3719_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_3720_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3721_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3722_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3723_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3724_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_3725_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_3726_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_3727_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_3728_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_3729_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_3730_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_3731_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_3732_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_3733_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_3734_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_3735_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_3736_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3737_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3738_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3739_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3740_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3741_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3742_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3743_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3744_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
                                   => ( ( 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 @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_3745_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_3746_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_3747_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3748_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_3749_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A2: $o,B2: $o] :
                ( A1
                = ( vEBT_Leaf @ A2 @ B2 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A22
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A22
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A22
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X7 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X3 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A22
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X7 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3750_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A4: $o,B4: $o] :
              ( A12
              = ( vEBT_Leaf @ A4 @ B4 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ( vEBT_invar_vebt @ X2 @ N ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                     => ( ( M5 = N )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N @ M5 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ( vEBT_invar_vebt @ X2 @ N ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                       => ( ( M5
                            = ( suc @ N ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N @ M5 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_invar_vebt @ X2 @ N ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                         => ( ( M5 = N )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N @ M5 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                & ! [X2: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X2 @ N )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X2 )
                                                      & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                           => ( vEBT_invar_vebt @ X2 @ N ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                           => ( ( M5
                                = ( suc @ N ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N @ M5 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                  & ! [X2: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X2 )
                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3751_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N3: nat,TreeList3: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X3 @ N3 ) ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) ) ) ).

% in_children_def
thf(fact_3752_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: 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 @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_3753_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ 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 @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3754_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ 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 @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3755_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ 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 @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_3756_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: 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 @ TreeList2 @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_3757_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M5: nat] :
              ( ( ( some_nat @ M5 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_3758_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ Info2 @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_3759_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_3760_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_3761_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_3762_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X3: nat,N3: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% high_def
thf(fact_3763_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_3764_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_3765_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_3766_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_3767_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_3768_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_3769_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_3770_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_3771_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_3772_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_3773_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_3774_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_3775_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_3776_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_3777_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_3778_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_3779_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_3780_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_3781_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_3782_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_3783_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_3784_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_3785_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_3786_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_3787_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_3788_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_3789_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3790_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3791_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3792_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_3793_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_3794_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_3795_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_3796_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_3797_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_3798_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_3799_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_3800_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_3801_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_3802_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_3803__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% \<open>deg div 2 = n\<close>
thf(fact_3804_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_3805_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3806_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3807_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3808_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3809_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3810_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3811_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3812_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3813_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3814_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3815_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3816_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3817_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3818_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3819_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_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_3829_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_3830_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_3831_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_3832_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_3833_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_3834_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_3835_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_3836_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_3837_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_3838_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_3839_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_3840_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_3841_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_3842_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_3843_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_3844_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_3845_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_3846_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_3847_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_3848_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_3849_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_3850_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_3851_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_3852_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_3853_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_3854_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_3855_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_3856_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_3857_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_3858_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = ( divide_divide_nat @ M @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_3859_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_3860_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_3861_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_3862_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_3863_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_3864_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_3865_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_3866_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_3867_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_3868_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_3869_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_3870_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_3871_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_3872_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_3873_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_3874_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_3875_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_3876_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_3877_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_3878_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_3879_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_3880_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_3881_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_3882_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_3883_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_3884_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_3885_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_3886_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_3887_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_3888_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_3889_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_3890_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_3891_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_3892_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_3893_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_3894_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_3895_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_3896_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_3897_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W2: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W2 ) )
      = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W2 ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_3898_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_3899_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% zle_add1_eq_le
thf(fact_3900_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% zle_diff1_eq
thf(fact_3901_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_3902_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_3903_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_3904_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_3905_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_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_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_3913_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_3914_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_3915_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_3916_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_3917_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_3918_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_3919_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_3920_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_3921_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_3922_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_3923_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_3924_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_3925_set__decode__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_3926_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_3927_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% nonpos_int_cases
thf(fact_3928_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_3929_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_3930_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% int_cases2
thf(fact_3931_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_3932_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_3933_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B5 )
       => ( ( ord_less_int @ zero_zero_int @ B5 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3934_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3935_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B5 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B5 )
             => ( ( ord_less_eq_int @ B5 @ B )
               => ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3936_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B5: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B5 )
             => ( ( ord_less_eq_int @ B5 @ B )
               => ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3937_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3938_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3939_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3940_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3941_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X2: int] :
              ( ( P @ X2 )
             => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3942_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_3943_odd__nonzero,axiom,
    ! [Z2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3944_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_3945_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_3946_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( semiri1314217659103216013at_int @ A2 )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_3947_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_3948_int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% int_int_eq
thf(fact_3949_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3950_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z2 )
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
thf(fact_3951_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_3952_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_3953_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z5: int] :
        ? [N3: nat] :
          ( Z5
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_3954_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X2: real] : ( member_real @ X2 @ S3 )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( member_real @ X4 @ S3 )
           => ( ord_less_eq_real @ X4 @ Z4 ) )
       => ? [Y3: real] :
            ( ! [X2: real] :
                ( ( member_real @ X2 @ S3 )
               => ( ord_less_eq_real @ X2 @ Y3 ) )
            & ! [Z4: real] :
                ( ! [X4: real] :
                    ( ( member_real @ X4 @ S3 )
                   => ( ord_less_eq_real @ X4 @ Z4 ) )
               => ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).

% complete_real
thf(fact_3955_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N ) ) ) ).

% nonneg_int_cases
thf(fact_3956_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_3957_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N ) ) ) ).

% zero_le_imp_eq_int
thf(fact_3958_imp__le__cong,axiom,
    ! [X: int,X8: int,P: $o,P4: $o] :
      ( ( X = X8 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
           => P4 ) ) ) ) ).

% imp_le_cong
thf(fact_3959_conj__le__cong,axiom,
    ! [X: int,X8: int,P: $o,P4: $o] :
      ( ( X = X8 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
            & P4 ) ) ) ) ).

% conj_le_cong
thf(fact_3960_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z2 )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% int_one_le_iff_zero_less
thf(fact_3961_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ Z2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).

% zless_imp_add1_zle
thf(fact_3962_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3963_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).

% le_imp_0_less
thf(fact_3964_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3965_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% add1_zle_eq
thf(fact_3966_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_3967_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M5: nat,N: nat] :
          ( Z2
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% int_diff_cases
thf(fact_3968_plusinfinity,axiom,
    ! [D: int,P4: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int,K2: int] :
            ( ( P4 @ X4 )
            = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( ( ord_less_int @ Z4 @ X4 )
             => ( ( P @ X4 )
                = ( P4 @ X4 ) ) )
         => ( ? [X_12: int] : ( P4 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_3969_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int,K2: int] :
            ( ( P1 @ X4 )
            = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( ( ord_less_int @ X4 @ Z4 )
             => ( ( P @ X4 )
                = ( P1 @ X4 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_3970_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X2: int] :
              ( ( P @ X2 )
             => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3971_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_3972_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_3973_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3974_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_3975_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_3976_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_3977_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_3978_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_3979_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_3980_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X3: real,Y2: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).

% minus_real_def
thf(fact_3981_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3982_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3983_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3984_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W2 ) @ ( times_times_rat @ Y @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_3985_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_3986_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_3987_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3988_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3989_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W2: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3990_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3991_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3992_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3993_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_3994_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_3995_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_3996_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_3997_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_3998_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_3999_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_4000_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_4001_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_4002_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_4003_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_4004_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_4005_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_4006_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_4007_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_4008_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_4009_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_4010_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_4011_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_4012_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_4013_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_4014_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_4015_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_4016_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_4017_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_4018_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_4019_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_4020_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_4021_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_4022_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_4023_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_4024_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_4025_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_4026_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_4027_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_4028_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_4029_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_4030_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_4031_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_4032_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_4033_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_4034_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_4035_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_4036_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_4037_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_4038_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_4039_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_4040_frac__eq__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ( times_times_rat @ X @ Z2 )
            = ( times_times_rat @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_4041_frac__eq__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W2 @ Z2 ) )
          = ( ( times_times_real @ X @ Z2 )
            = ( times_times_real @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_4042_frac__eq__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W2 @ Z2 ) )
          = ( ( times_times_complex @ X @ Z2 )
            = ( times_times_complex @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_4043_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_4044_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_4045_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_4046_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_4047_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_4048_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_4049_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_4050_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_4051_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_4052_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_4053_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_4054_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B )
       => ( A
          = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_4055_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_4056_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_4057_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B @ C )
          = A )
        = ( B
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_4058_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_4059_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_4060_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_4061_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_4062_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_4063_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_4064_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_4065_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_4066_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_4067_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_4068_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_4069_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_4070_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_4071_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_4072_power__one__over,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_4073_power__one__over,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_4074_power__one__over,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_4075_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z2: 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 @ Z2 )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_4076_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z2: 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 @ Z2 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_4077_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z2: 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 @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_4078_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z2: 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 @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_4079_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z2: 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 @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_4080_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z2: 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 @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_4081_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_4082_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_4083_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_4084_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_4085_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_4086_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_4087_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_4088_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_4089_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_4090_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_4091_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_4092_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_4093_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_4094_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_4095_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4096_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4097_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_4098_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_4099_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_4100_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_4101_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_4102_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_4103_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_4104_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_4105_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_4106_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_4107_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_4108_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_4109_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_4110_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_4111_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
       => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_4112_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
       => ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_4113_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_4114_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_4115_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_4116_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_4117_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_4118_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_4119_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_4120_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_4121_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W2 ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_4122_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_4123_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_4124_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W2 )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_4125_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_4126_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W2 )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_4127_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_4128_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_4129_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_4130_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_4131_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_4132_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_4133_add__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_4134_add__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_4135_add__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_4136_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_4137_add__frac__num,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_4138_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_4139_add__num__frac,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_4140_add__num__frac,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_4141_add__num__frac,axiom,
    ! [Y: complex,Z2: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_4142_add__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_4143_add__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_4144_add__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_4145_divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_4146_divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_4147_divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_4148_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4149_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4150_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4151_diff__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4152_diff__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4153_diff__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4154_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_4155_diff__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_4156_diff__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_4157_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_4158_divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_4159_divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_4160_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_4161_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_4162_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_4163_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_4164_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4165_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4166_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4167_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4168_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4169_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4170_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4171_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4172_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4173_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4174_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4175_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4176_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_4177_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_4178_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_4179_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_4180_two__realpow__ge__one,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% two_realpow_ge_one
thf(fact_4181_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: real,A: real,B: real] :
      ( ( nO_MATCH_nat_real @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4182_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4183_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4184_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4185_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: real,A: real,B: real] :
      ( ( nO_MATCH_int_real @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4186_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4187_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4188_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4189_distrib__right__NO__MATCH,axiom,
    ! [X: real,Y: real,C: real,A: real,B: real] :
      ( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ C )
     => ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4190_distrib__right__NO__MATCH,axiom,
    ! [X: real,Y: real,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_real_rat @ ( divide_divide_real @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_4191_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: real,B: real,C: real] :
      ( ( nO_MATCH_nat_real @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4192_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4193_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4194_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4195_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: real,B: real,C: real] :
      ( ( nO_MATCH_int_real @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4196_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4197_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4198_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4199_distrib__left__NO__MATCH,axiom,
    ! [X: real,Y: real,A: real,B: real,C: real] :
      ( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4200_distrib__left__NO__MATCH,axiom,
    ! [X: real,Y: real,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_real_rat @ ( divide_divide_real @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_4201_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: real,B: real,C: real] :
      ( ( nO_MATCH_nat_real @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4202_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4203_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4204_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: real,B: real,C: real] :
      ( ( nO_MATCH_int_real @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4205_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4206_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4207_right__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: real,B: real,C: real] :
      ( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4208_right__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_real_rat @ ( divide_divide_real @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4209_right__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: int,B: int,C: int] :
      ( ( nO_MATCH_real_int @ ( divide_divide_real @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4210_right__diff__distrib__NO__MATCH,axiom,
    ! [X: complex,Y: complex,A: real,B: real,C: real] :
      ( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ A )
     => ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_4211_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: real,A: real,B: real] :
      ( ( nO_MATCH_nat_real @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4212_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4213_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4214_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: real,A: real,B: real] :
      ( ( nO_MATCH_int_real @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4215_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4216_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4217_left__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,C: real,A: real,B: real] :
      ( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ C )
     => ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4218_left__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_real_rat @ ( divide_divide_real @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4219_left__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,C: int,A: int,B: int] :
      ( ( nO_MATCH_real_int @ ( divide_divide_real @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4220_left__diff__distrib__NO__MATCH,axiom,
    ! [X: complex,Y: complex,C: real,A: real,B: real] :
      ( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ C )
     => ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_4221_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_4222_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_4223_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
       => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_4224_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
       => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_4225_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_4226_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_4227_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_4228_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_4229_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_4230_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_4231_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_4232_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_4233_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_4234_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_4235_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_4236_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_4237_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_4238_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_4239_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_4240_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_4241_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_4242_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_4243_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_4244_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_4245_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_4246_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_4247_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_4248_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_4249_frac__le__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_4250_frac__le__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_4251_frac__less__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_4252_frac__less__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_4253_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4254_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4255_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4256_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4257_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4258_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4259_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4260_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4261_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4262_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4263_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4264_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4265_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4266_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4267_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4268_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4269_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4270_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4271_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4272_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4273_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4274_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4275_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4276_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4277_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_4278_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_4279_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4280_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4281_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4282_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4283_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4284_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4285_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4286_minus__divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4287_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4288_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4289_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4290_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4291_power__diff,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4292_power__diff,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4293_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_4294_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_4295_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_4296_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_4297_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_4298_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4299_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4300_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4301_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4302_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4303_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4304_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4305_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4306_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4307_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4308_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4309_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4310_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_4311_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_4312_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_4313_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_4314_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_4315_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_4316_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_4317_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_4318_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_4319_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_4320_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_4321_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_4322_nat__approx__posE,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ~ ! [N: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_4323_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E2 )
     => ~ ! [N: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_4324_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_4325_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_4326_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_4327_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_4328_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_4329_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_4330_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_4331_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_4332_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_4333_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_4334_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_4335_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_4336_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_4337_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_4338_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_4339_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( N2
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_4340_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_4341_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_4342_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_4343_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_4344_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_4345_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_4346_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_4347_zdiv__numeral__Bit0,axiom,
    ! [V: num,W2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_4348_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_4349_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_4350_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_4351_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_4352_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_4353_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_4354_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_4355_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_4356_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_4357_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_4358_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_4359_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_4360_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_4361_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_4362_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_4363_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_4364_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_4365_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_4366_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_4367_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_4368_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_4369_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_4370_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_4371_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_4372_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_4373_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_4374_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_4375_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_4376_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_4377_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( ( ord_less_eq_int @ A @ A5 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_4378_zdiv__mono2,axiom,
    ! [A: int,B5: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B5 )
       => ( ( ord_less_eq_int @ B5 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_4379_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_4380_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( ( ord_less_eq_int @ A @ A5 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_4381_zdiv__mono2__neg,axiom,
    ! [A: int,B5: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B5 )
       => ( ( ord_less_eq_int @ B5 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_4382_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_4383_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_4384_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_4385_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_4386_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_4387_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_4388_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_4389_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_4390_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_4391_real__of__nat__div4,axiom,
    ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_4392_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A3 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_4393_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_4394_real__of__nat__div2,axiom,
    ! [N2: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_4395_real__of__nat__div3,axiom,
    ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_4396_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_4397_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_pos_eq
thf(fact_4398_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_neg_eq
thf(fact_4399_split__zdiv,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( divide_divide_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_4400_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_4401_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_4402_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_4403_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_4404_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_4405_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_4406_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_4407_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_4408_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_4409_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_4410_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_4411_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_4412_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_4413_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_4414_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_4415_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_4416_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_4417_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_4418_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_4419_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_4420_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N2 )
          = M )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_4421_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N3: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M6 @ N3 )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).

% div_if
thf(fact_4422_div__nat__eqI,axiom,
    ! [N2: nat,Q4: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q4 ) ) ) ).

% div_nat_eqI
thf(fact_4423_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q4 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_4424_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_4425_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_4426_split__div,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_4427_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_4428_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_4429_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_4430_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_4431_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4432_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4433_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_4434_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_4435_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4436_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4437_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_4438_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q6: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q6 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q6 ) ) )
            & ( P @ Q6 ) ) ) ) ).

% split_div'
thf(fact_4439_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_4440_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_4441_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_4442_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_4443_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_4444_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_4445_inrange,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_4446_linear__plus__1__le__power,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).

% linear_plus_1_le_power
thf(fact_4447_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_4448_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_4449_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_4450_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_4451_i0__less,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_4452_Icc__eq__Icc,axiom,
    ! [L: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H2 )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H2 )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4453_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_4454_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_4455_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_4456_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_4457_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_4458_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4459_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4460_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4461_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4462_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4463_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4464_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_4465_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_4466_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_4467_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( set_or5403411693681687835d_enat @ A @ B )
        = bot_bo7653980558646680370d_enat )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4468_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4469_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_4470_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_4471_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_4472_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_4473_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_4474_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat
        = ( set_or5403411693681687835d_enat @ A @ B ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4475_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4476_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_4477_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_4478_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_4479_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_4480_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_4481_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4482_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4483_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4484_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4485_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4486_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4487_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( set_or5403411693681687835d_enat @ A @ B )
        = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
thf(fact_4488_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_4489_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_4490_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_4491_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_4492_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_4493_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_4494_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_4495_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4496_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4497_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4498_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4499_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4500_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4501_set__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4502_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4503_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4504_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4505_not__iless0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_4506_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_4507_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).

% i0_lb
thf(fact_4508_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N2: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% enat_less_induct
thf(fact_4509_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).

% enat_0_less_mult_iff
thf(fact_4510_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_4511_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_4512_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
            & ( P @ M6 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less
thf(fact_4513_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
           => ( P @ M6 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less
thf(fact_4514_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A3: set_complex,X: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
     => ( ( member_complex @ X @ A3 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4515_set__update__subsetI,axiom,
    ! [Xs2: list_real,A3: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A3 )
     => ( ( member_real @ X @ A3 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4516_set__update__subsetI,axiom,
    ! [Xs2: list_set_nat,A3: set_set_nat,X: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A3 )
     => ( ( member_set_nat @ X @ A3 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4517_set__update__subsetI,axiom,
    ! [Xs2: list_int,A3: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
     => ( ( member_int @ X @ A3 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4518_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A3: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
     => ( ( member_VEBT_VEBT @ X @ A3 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4519_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A3: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
     => ( ( member_nat @ X @ A3 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_4520_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4521_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4522_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4523_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4524_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4525_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4526_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_complex,X: complex] :
      ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4527_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4528_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4529_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4530_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4531_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4532_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_4533_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4534_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4535_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4536_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4537_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J )
        = ( ( ( I = J )
           => X )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4538_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4539_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_4540_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A4: real,B4: real,C3: real] :
            ( ( P @ A4 @ B4 )
           => ( ( P @ B4 @ C3 )
             => ( ( ord_less_eq_real @ A4 @ B4 )
               => ( ( ord_less_eq_real @ B4 @ C3 )
                 => ( P @ A4 @ C3 ) ) ) ) )
       => ( ! [X4: real] :
              ( ( ord_less_eq_real @ A @ X4 )
             => ( ( ord_less_eq_real @ X4 @ B )
               => ? [D3: real] :
                    ( ( ord_less_real @ zero_zero_real @ D3 )
                    & ! [A4: real,B4: real] :
                        ( ( ( ord_less_eq_real @ A4 @ X4 )
                          & ( ord_less_eq_real @ X4 @ B4 )
                          & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
                       => ( P @ A4 @ B4 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_4541_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: 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 @ TreeList2 ) )
     => ( ( 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 @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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_4542_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: 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 @ TreeList2 ) )
     => ( ( 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 @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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_4543_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_4544_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_4545_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_4546_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_4547_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_4548_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_4549_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_4550_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_4551_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = N2 ) ).

% idiff_0_right
thf(fact_4552_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_4553_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_4554_succ__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ X @ Y2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_4555_pred__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ Y2 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_4556_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_4557_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_4558_max__bot,axiom,
    ! [X: set_Extended_enat] :
      ( ( ord_ma4205026669011143323d_enat @ bot_bo7653980558646680370d_enat @ X )
      = X ) ).

% max_bot
thf(fact_4559_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_4560_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_4561_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_4562_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_4563_max__bot,axiom,
    ! [X: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X )
      = X ) ).

% max_bot
thf(fact_4564_max__bot2,axiom,
    ! [X: set_Extended_enat] :
      ( ( ord_ma4205026669011143323d_enat @ X @ bot_bo7653980558646680370d_enat )
      = X ) ).

% max_bot2
thf(fact_4565_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_4566_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_4567_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_4568_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_4569_max__bot2,axiom,
    ! [X: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ X @ bot_bo4199563552545308370d_enat )
      = X ) ).

% max_bot2
thf(fact_4570_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_4571_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_4572_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_4573_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_4574_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_4575_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_4576_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_4577_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_4578_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_4579_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_4580_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_4581_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4582_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_4583_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_4584_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_4585_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_4586_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_4587_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_4588_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(4)
thf(fact_4589_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_4590_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_4591_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_4592_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_4593_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_4594_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(3)
thf(fact_4595_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_4596_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_4597_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_4598_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_4599_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_4600_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_4601_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_4602_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_4603_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_4604_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_4605_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_4606_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_4607_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_4608_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_4609_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_4610_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_4611_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_4612_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(6)
thf(fact_4613_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_4614_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_4615_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_4616_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_4617_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_4618_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(5)
thf(fact_4619_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_4620_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_4621_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_4622_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_4623_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_4624_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_4625_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_4626_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_4627_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_4628_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_4629_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_4630_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_4631_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_4632_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_4633_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_4634_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_4635_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_4636_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_4637_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_4638_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_4639_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_4640_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_4641_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_4642_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_4643_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_4644_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4645_max__def__raw,axiom,
    ( ord_max_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4646_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4647_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4648_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4649_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4650_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4651_iadd__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_4652_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_4653_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A7: set_complex,B7: set_complex] :
          ( ord_less_complex_o
          @ ^ [X3: complex] : ( member_complex @ X3 @ A7 )
          @ ^ [X3: complex] : ( member_complex @ X3 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4654_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A7: set_real,B7: set_real] :
          ( ord_less_real_o
          @ ^ [X3: real] : ( member_real @ X3 @ A7 )
          @ ^ [X3: real] : ( member_real @ X3 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4655_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A7: set_set_nat,B7: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A7 )
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4656_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ord_less_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A7 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4657_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( ord_less_int_o
          @ ^ [X3: int] : ( member_int @ X3 @ A7 )
          @ ^ [X3: int] : ( member_int @ X3 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4658_imult__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_4659_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y ) ) ) ).

% of_nat_max
thf(fact_4660_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).

% of_nat_max
thf(fact_4661_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_4662_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y ) ) ) ).

% of_nat_max
thf(fact_4663_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_4664_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_4665_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_4666_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_4667_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_4668_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_4669_lambda__one,axiom,
    ( ( ^ [X3: complex] : X3 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_4670_lambda__one,axiom,
    ( ( ^ [X3: real] : X3 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_4671_lambda__one,axiom,
    ( ( ^ [X3: rat] : X3 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_4672_lambda__one,axiom,
    ( ( ^ [X3: nat] : X3 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_4673_lambda__one,axiom,
    ( ( ^ [X3: int] : X3 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_4674_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_4675_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_4676_max__absorb2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_ma741700101516333627d_enat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4677_max__absorb2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_max_Code_integer @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4678_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_max_set_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4679_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4680_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4681_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4682_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4683_max__absorb1,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ( ord_ma741700101516333627d_enat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4684_max__absorb1,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ( ord_max_Code_integer @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4685_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_max_set_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4686_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4687_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4688_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4689_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4690_max__def,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4691_max__def,axiom,
    ( ord_max_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4692_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4693_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4694_max__def,axiom,
    ( ord_max_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4695_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4696_max__def,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_4697_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4698_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z2 ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4699_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4700_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z2 ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4701_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z2 ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4702_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4703_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4704_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4705_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4706_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z2 )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z2 ) @ ( plus_p5714425477246183910nteger @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4707_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z2 )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z2 ) @ ( minus_8373710615458151222nteger @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4708_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4709_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4710_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4711_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).

% nat_add_max_right
thf(fact_4712_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N2 @ Q4 ) ) ) ).

% nat_add_max_left
thf(fact_4713_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).

% nat_mult_max_right
thf(fact_4714_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).

% nat_mult_max_left
thf(fact_4715_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_4716_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_4717_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_4718_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_4719_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_4720_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_4721_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_4722_finite__int__segment,axiom,
    ! [A: real,B: real] :
      ( finite_finite_real
      @ ( collect_real
        @ ^ [X3: real] :
            ( ( member_real @ X3 @ ring_1_Ints_real )
            & ( ord_less_eq_real @ A @ X3 )
            & ( ord_less_eq_real @ X3 @ B ) ) ) ) ).

% finite_int_segment
thf(fact_4723_finite__int__segment,axiom,
    ! [A: rat,B: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [X3: rat] :
            ( ( member_rat @ X3 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ A @ X3 )
            & ( ord_less_eq_rat @ X3 @ B ) ) ) ) ).

% finite_int_segment
thf(fact_4724_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_4725_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4726_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
      = ( ord_max_nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_4727_finite__lists__length__eq,axiom,
    ! [A3: set_int,N2: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
              & ( ( size_size_list_int @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4728_finite__lists__length__eq,axiom,
    ! [A3: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A3 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4729_finite__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A3 )
              & ( ( size_s5460976970255530739at_nat @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4730_finite__lists__length__eq,axiom,
    ! [A3: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs ) @ A3 )
              & ( ( size_s3941691890525107288d_enat @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4731_finite__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4732_finite__lists__length__eq,axiom,
    ! [A3: set_o,N2: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
              & ( ( size_size_list_o @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4733_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
              & ( ( size_size_list_nat @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4734_finite__lists__length__le,axiom,
    ! [A3: set_int,N2: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4735_finite__lists__length__le,axiom,
    ! [A3: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4736_finite__lists__length__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4737_finite__lists__length__le,axiom,
    ! [A3: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4738_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4739_finite__lists__length__le,axiom,
    ! [A3: set_o,N2: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4740_finite__lists__length__le,axiom,
    ! [A3: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4741_aset_I2_J,axiom,
    ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D4 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
             => ( ( P @ ( plus_plus_int @ X2 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_4742_aset_I1_J,axiom,
    ! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D4 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
             => ( ( P @ ( plus_plus_int @ X2 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_4743_bset_I2_J,axiom,
    ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D4 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
             => ( ( P @ ( minus_minus_int @ X2 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_4744_bset_I1_J,axiom,
    ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D4 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
             => ( ( P @ ( minus_minus_int @ X2 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_4745_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int,K2: int] :
            ( ( P @ X4 )
            = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X7: int] : ( P @ X7 ) )
          = ( ? [X3: int] :
                ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X3 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_4746_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X2 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_4747_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X2 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_4748_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 != T )
             => ( ( plus_plus_int @ X2 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_4749_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 = T )
             => ( ( plus_plus_int @ X2 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_4750_bset_I7_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X2 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_4751_bset_I5_J,axiom,
    ! [D4: int,B3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X2 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_4752_bset_I4_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 != T )
             => ( ( minus_minus_int @ X2 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_4753_bset_I3_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 = T )
             => ( ( minus_minus_int @ X2 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_4754_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( 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 @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_4755_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_4756_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X2 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_4757_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X2 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_4758_bset_I8_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X2 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_4759_bset_I6_J,axiom,
    ! [D4: int,B3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X2 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_4760_cpmi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( P @ X4 )
              = ( P4 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ B3 )
                     => ( X4
                       != ( plus_plus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( minus_minus_int @ X4 @ D4 ) ) ) )
         => ( ! [X4: int,K2: int] :
                ( ( P4 @ X4 )
                = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X7: int] : ( P @ X7 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ B3 )
                        & ( P @ ( plus_plus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_4761_cppi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( P @ X4 )
              = ( P4 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ A3 )
                     => ( X4
                       != ( minus_minus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( plus_plus_int @ X4 @ D4 ) ) ) )
         => ( ! [X4: int,K2: int] :
                ( ( P4 @ X4 )
                = ( P4 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X7: int] : ( P @ X7 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ A3 )
                        & ( P @ ( minus_minus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_4762_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_4763_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B4 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A4 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) )
             => ( Y
               != ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
               => ( Y
                 != ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                          & ~ ( ( 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 @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_4764_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_4765_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ 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 @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_4766_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A4 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B4 )
                  & ( 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,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_4767_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A4 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B4 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_4768_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( 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,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_4769_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,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ TreeList ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_4770_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A4 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B4 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_4771_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,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_4772_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,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ TreeList ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_4773_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A4 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B4 )
                  & ( 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,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_4774_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( 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,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_4775_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_4776_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_4777_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_4778_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B4 ) ) ) )
       => ( ! [A4: $o] :
              ( ? [B4: $o] :
                  ( X
                  = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A4 @ $false ) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A4 @ B4 ) )
               => ( ? [N: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A4 @ B4 ) ) ) )
           => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_4779_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu2: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B4 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B4
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B4
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N: nat] :
                  ( Xa2
                  = ( suc @ N ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_4780_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A4
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A4
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A4 @ B4 ) )
               => ( ? [Va: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B4
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B4
                       => ( ( A4
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A4
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_4781_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_4782_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z5: real] :
              ( ( power_power_real @ Z5 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_4783_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_4784_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B4 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B4
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B4
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N: nat] :
                    ( ( Xa2
                      = ( suc @ N ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_4785_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A4
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A4
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A4 @ B4 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( B4
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B4
                           => ( ( A4
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A4
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_4786_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4787_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4788_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4789_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4790_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4791_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4792_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4793_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4794_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4795_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4796_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4797_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4798_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4799_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4800_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z2 )
      = ( ( ord_le72135733267957522d_enat @ X @ Z2 )
        & ( ord_le72135733267957522d_enat @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4801_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z2 )
      = ( ( ord_le6747313008572928689nteger @ X @ Z2 )
        & ( ord_le6747313008572928689nteger @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4802_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ( ord_less_real @ X @ Z2 )
        & ( ord_less_real @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4803_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
      = ( ( ord_less_rat @ X @ Z2 )
        & ( ord_less_rat @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4804_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z2 )
      = ( ( ord_less_num @ X @ Z2 )
        & ( ord_less_num @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4805_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
      = ( ( ord_less_nat @ X @ Z2 )
        & ( ord_less_nat @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4806_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z2 )
      = ( ( ord_less_int @ X @ Z2 )
        & ( ord_less_int @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4807_max__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q4 @ zero_z5237406670263579293d_enat )
      = Q4 ) ).

% max_enat_simps(2)
thf(fact_4808_max__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q4 )
      = Q4 ) ).

% max_enat_simps(3)
thf(fact_4809_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
      = ( ( ord_le2932123472753598470d_enat @ B @ A )
        & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4810_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
      = ( ( ord_le3102999989581377725nteger @ B @ A )
        & ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4811_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4812_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4813_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4814_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4815_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4816_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4817_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4818_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4819_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4820_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4821_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4822_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4823_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4824_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4825_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4826_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4827_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_4828_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ( ord_le2932123472753598470d_enat @ D @ B )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4829_max_Omono,axiom,
    ! [C: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A )
     => ( ( ord_le3102999989581377725nteger @ D @ B )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C @ D ) @ ( ord_max_Code_integer @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4830_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4831_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4832_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4833_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4834_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4835_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( A
        = ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.orderE
thf(fact_4836_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4837_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_4838_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4839_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_4840_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_4841_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( ord_max_Code_integer @ A @ B ) )
     => ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% max.orderI
thf(fact_4842_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_4843_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_4844_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_4845_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_4846_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4847_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le3102999989581377725nteger @ B @ A )
         => ~ ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4848_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4849_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4850_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4851_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4852_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ A )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4853_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ A )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4854_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4855_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4856_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4857_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4858_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( A2
          = ( ord_ma741700101516333627d_enat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4859_max_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( A2
          = ( ord_max_Code_integer @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4860_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4861_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( A2
          = ( ord_max_num @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4862_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4863_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4864_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4865_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded1
thf(fact_4866_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4867_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_4868_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4869_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_4870_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4871_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : ( ord_le3102999989581377725nteger @ B @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded2
thf(fact_4872_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4873_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_4874_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4875_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_4876_le__max__iff__disj,axiom,
    ! [Z2: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z2 @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le2932123472753598470d_enat @ Z2 @ X )
        | ( ord_le2932123472753598470d_enat @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4877_le__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Z2 @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le3102999989581377725nteger @ Z2 @ X )
        | ( ord_le3102999989581377725nteger @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4878_le__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z2 @ X )
        | ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4879_le__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z2 @ X )
        | ( ord_less_eq_num @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4880_le__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z2 @ X )
        | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4881_le__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z2 @ X )
        | ( ord_less_eq_int @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4882_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4883_max_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_max_Code_integer @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4884_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4885_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4886_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4887_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4888_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4889_max_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_max_Code_integer @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4890_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4891_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4892_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4893_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4894_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4895_max_OcoboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4896_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4897_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4898_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4899_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4900_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4901_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ B )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4902_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4903_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4904_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4905_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4906_less__max__iff__disj,axiom,
    ! [Z2: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z2 @ X )
        | ( ord_le72135733267957522d_enat @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4907_less__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z2 @ X )
        | ( ord_le6747313008572928689nteger @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4908_less__max__iff__disj,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z2 @ X )
        | ( ord_less_real @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4909_less__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z2 @ X )
        | ( ord_less_rat @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4910_less__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y: num] :
      ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z2 @ X )
        | ( ord_less_num @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4911_less__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z2 @ X )
        | ( ord_less_nat @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4912_less__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z2 @ X )
        | ( ord_less_int @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4913_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4914_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4915_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4916_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4917_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4918_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4919_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4920_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( ( A2
            = ( ord_ma741700101516333627d_enat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4921_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( A2
            = ( ord_max_Code_integer @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4922_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( A2
            = ( ord_max_real @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4923_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( ord_max_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4924_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( A2
            = ( ord_max_num @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4925_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( ord_max_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4926_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( ord_max_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4927_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4928_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4929_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4930_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4931_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4932_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4933_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_4934_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4935_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4936_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4937_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4938_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4939_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4940_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_4941_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B4 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A4 @ B4 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A4 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A4 @ B4 ) )
                 => ! [N: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A4 @ B4 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_4942_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B4 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A4 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                              & ~ ( ( 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 @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4943_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ 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,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_4944_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A4 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B4 )
                      & ( 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,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_4945_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ 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,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_4946_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_4947_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A4 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B4 )
                      & ( 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,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S2 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_4948_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 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A4 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4949_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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ TreeList ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4950_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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4951_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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ TreeList ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4952_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_4953_lemma__termdiff3,axiom,
    ! [H2: real,Z2: real,K5: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4954_lemma__termdiff3,axiom,
    ! [H2: complex,Z2: complex,K5: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4955_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_4956_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_4957_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_4958_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_4959_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_4960_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_4961_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_4962_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_4963_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_4964_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_4965_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_4966_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_4967_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_4968_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_4969_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_4970_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_4971_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_4972_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_4973_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_4974_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_4975_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_4976_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_4977_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_4978_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_4979_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_4980_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_4981_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_4982_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_4983_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_4984_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_4985_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_4986_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_4987_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_4988_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_4989_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_4990_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_4991_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_4992_ln__realpow,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_4993_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_4994_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_4995_norm__not__less__zero,axiom,
    ! [X: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_4996_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_4997_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_4998_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_4999_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_5000_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_5001_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_5002_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N2: nat,Z2: real] :
      ( ( ( power_power_real @ W2 @ N2 )
        = ( power_power_real @ Z2 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_5003_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N2: nat,Z2: complex] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = ( power_power_complex @ Z2 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_5004_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_5005_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_5006_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_5007_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_5008_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_5009_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_5010_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_5011_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_5012_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_5013_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_5014_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_5015_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_5016_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_5017_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_5018_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z2: 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 @ Z2 ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_5019_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_5020_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_5021_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_5022_power__eq__1__iff,axiom,
    ! [W2: real,N2: nat] :
      ( ( ( power_power_real @ W2 @ N2 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W2 )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5023_power__eq__1__iff,axiom,
    ! [W2: complex,N2: nat] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W2 )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5024_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5025_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5026_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_5027_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
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_5028_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_5029_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5030_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5031_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5032_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5033_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5034_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5035_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5036_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X3: nat,N3: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% low_def
thf(fact_5037_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_5038_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_5039_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_5040_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_5041_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_5042_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_5043_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_5044_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_5045_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_5046_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_5047_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_5048_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_5049_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_5050_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_5051_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_5052_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_5053_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_5054_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_5055_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_5056_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_5057_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_5058_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_5059_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_5060_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_5061_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_5062_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_5063_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_5064_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_5065_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_5066_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_5067_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_5068_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_5069_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_5070_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_5071_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_5072_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_5073_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_numeral
thf(fact_5074_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_numeral
thf(fact_5075_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_numeral
thf(fact_5076_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_numeral
thf(fact_5077_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5078_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5079_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5080_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5081_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_5082_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_5083_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_5084_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_5085_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_5086_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_5087_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_5088_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_5089_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_5090_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_5091_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_5092_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_5093_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_5094_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_5095_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_5096_abs__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_divide
thf(fact_5097_abs__divide,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_divide
thf(fact_5098_abs__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_divide
thf(fact_5099_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_5100_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_5101_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_5102_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_5103_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_5104_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_5105_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_5106_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_5107_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_5108_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_5109_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_5110_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_5111_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_5112_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_5113_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_5114_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_5115_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_5116_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_5117_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_5118_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_5119_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_5120_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_5121_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_5122_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_5123_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_5124_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5125_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5126_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5127_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5128_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_5129_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_5130_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_5131_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_5132_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_5133_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_5134_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_5135_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_5136_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_5137_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_5138_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_5139_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_5140_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_5141_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_5142_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_5143_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_5144_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_5145_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_5146_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_5147_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_5148_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_5149_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_5150_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_5151_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_5152_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_5153_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_5154_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_5155_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_5156_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_5157_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_5158_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_5159_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_5160_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_5161_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_5162_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5163_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5164_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5165_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5166_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_5167_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_5168_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_5169_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_5170_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_5171_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_5172_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_5173_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_5174_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5175_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5176_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5177_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5178_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5179_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_5180_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_5181_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_5182_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_5183_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5184_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5185_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5186_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5187_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5188_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_5189_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_5190_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_5191_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_5192_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_5193_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_5194_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_5195_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_5196_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_5197_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_5198_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_5199_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_5200_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5201_zero__less__power__abs__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
      = ( ( A != zero_zero_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5202_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
      = ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5203_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
      = ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5204_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_5205_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_5206_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_5207_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_5208_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_5209_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_5210_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_5211_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_5212_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_5213_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_5214_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_5215_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_5216_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_5217_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_5218_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_5219_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_5220_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_5221_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_5222_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_5223_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_5224_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_5225_unique__remainder,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_5226_unique__quotient,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( Q4 = Q5 ) ) ) ).

% unique_quotient
thf(fact_5227_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_5228_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_5229_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_5230_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_5231_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_5232_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5233_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5234_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5235_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_5236_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_5237_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_5238_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_5239_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_5240_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_5241_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_5242_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_5243_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5244_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_5245_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_5246_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_5247_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5248_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5249_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5250_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5251_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5252_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5253_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5254_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5255_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_5256_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_5257_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_5258_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B5 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_5259_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B5: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B5 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B5 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_5260_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B5 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B5 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_5261_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_5262_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_5263_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_5264_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_5265_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_5266_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_5267_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_5268_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_5269_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_5270_Ints__abs,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).

% Ints_abs
thf(fact_5271_Ints__abs,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( member_rat @ ( abs_abs_rat @ A ) @ ring_1_Ints_rat ) ) ).

% Ints_abs
thf(fact_5272_Ints__abs,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( member_Code_integer @ ( abs_abs_Code_integer @ A ) @ ring_11222124179247155820nteger ) ) ).

% Ints_abs
thf(fact_5273_Ints__abs,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).

% Ints_abs
thf(fact_5274_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_5275_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q4 ) ) ).

% div_int_unique
thf(fact_5276_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5277_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_5278_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5279_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5280_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5281_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_5282_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5283_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5284_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5285_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5286_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5287_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5288_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5289_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5290_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5291_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5292_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5293_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5294_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5295_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5296_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5297_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5298_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5299_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5300_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5301_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5302_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5303_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5304_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5305_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5306_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5307_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5308_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_5309_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_5310_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_5311_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_5312_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_5313_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_5314_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5315_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5316_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5317_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5318_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5319_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5320_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5321_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5322_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5323_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5324_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5325_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5326_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5327_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5328_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5329_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5330_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5331_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_5332_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_5333_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_5334_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_5335_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5336_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_5337_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_5338_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_5339_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_5340_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_5341_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_5342_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_5343_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_5344_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_5345_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D5: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_5346_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D5: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_5347_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_5348_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_5349_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_5350_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
          = N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
         != N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_5351_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P6: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P6 )
       => ( ( ord_less_nat @ M @ P6 )
         => ( ! [N: nat] :
                ( ( ord_less_nat @ N @ P6 )
               => ( ( P @ N )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N ) @ P6 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_5352_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
     => ( ! [M5: nat,N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ( P @ N @ ( modulo_modulo_nat @ M5 @ N ) )
             => ( P @ M5 @ N ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_5353_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_5354_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_5355_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q2: nat] :
          ( M
          = ( times_times_nat @ D @ Q2 ) ) ) ).

% mod_eq_0D
thf(fact_5356_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_5357_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N3 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_5358_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_5359_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_5360_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_5361_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_5362_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_5363_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_5364_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_5365_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5366_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5367_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5368_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5369_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5370_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5371_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5372_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5373_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_5374_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5375_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_5376_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_5377_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5378_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5379_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5380_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5381_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5382_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5383_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5384_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5385_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_5386_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_5387_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5388_zero__le__power__abs,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5389_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5390_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_5391_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_5392_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_5393_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_5394_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_5395_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_5396_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_5397_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_5398_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_5399_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_5400_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_5401_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_5402_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_5403_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5404_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5405_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5406_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5407_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5408_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5409_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5410_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5411_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5412_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5413_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5414_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5415_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5416_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5417_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5418_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5419_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5420_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_5421_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_5422_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5423_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_5424_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_5425_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5426_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5427_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5428_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_5429_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_5430_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_5431_cong__exp__iff__simps_I6_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_5432_cong__exp__iff__simps_I6_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_5433_cong__exp__iff__simps_I6_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_5434_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q4: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_5435_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q4: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_5436_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q4: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_5437_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_5438_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_5439_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5440_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_5441_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_5442_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5443_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_5444_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_5445_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5446_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_5447_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_5448_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5449_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_5450_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_5451_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5452_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5453_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5454_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5455_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5456_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5457_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5458_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5459_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5460_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5461_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_5462_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_5463_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5464_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_5465_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_5466_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5467_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_5468_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_5469_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5470_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_5471_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_5472_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5473_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_5474_div__less__mono,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A3 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B3 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_5475_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_real_def
thf(fact_5476_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q4 )
        = ( modulo_modulo_nat @ N2 @ Q4 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_5477_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q4 )
        = ( modulo_modulo_nat @ N2 @ Q4 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S2: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_5478_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q2: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_5479_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q4 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5480_div__mod__decomp,axiom,
    ! [A3: nat,N2: nat] :
      ( A3
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A3 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_5481_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q4 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_5482_finite__abs__int__segment,axiom,
    ! [A: real] :
      ( finite_finite_real
      @ ( collect_real
        @ ^ [K3: real] :
            ( ( member_real @ K3 @ ring_1_Ints_real )
            & ( ord_less_eq_real @ ( abs_abs_real @ K3 ) @ A ) ) ) ) ).

% finite_abs_int_segment
thf(fact_5483_finite__abs__int__segment,axiom,
    ! [A: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [K3: rat] :
            ( ( member_rat @ K3 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A ) ) ) ) ).

% finite_abs_int_segment
thf(fact_5484_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5485_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5486_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5487_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5488_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( X != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5489_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( ( member_real @ X @ ring_1_Ints_real )
     => ( ( X != zero_zero_real )
       => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5490_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( X != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5491_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( X != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_5492_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5493_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5494_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5495_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5496_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( ( member_real @ X @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5497_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5498_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_5499_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
       => ( ( X = Y )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5500_Ints__eq__abs__less1,axiom,
    ! [X: real,Y: real] :
      ( ( member_real @ X @ ring_1_Ints_real )
     => ( ( member_real @ Y @ ring_1_Ints_real )
       => ( ( X = Y )
          = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5501_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( member_rat @ Y @ ring_1_Ints_rat )
       => ( ( X = Y )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5502_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( member_int @ Y @ ring_1_Ints_int )
       => ( ( X = Y )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_5503_split__mod,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_5504_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_5505_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5506_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5507_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5508_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5509_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_5510_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_5511_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_5512_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_5513_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5514_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5515_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5516_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_5517_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X4: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
         => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5518_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5519_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
         => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5520_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5521_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5522_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( 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 @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5523_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( 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 @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5524_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( 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 @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5525_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_5526_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_5527_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_5528_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_5529_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_5530_abs__square__less__1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_5531_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_5532_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_5533_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_5534_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_5535_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5536_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_5537_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_5538_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_5539_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5540_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5541_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5542_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5543_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_5544_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_5545_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_5546_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5547_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5548_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5549_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5550_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ 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 )
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5551_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5552_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5553_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5554_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5555_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_5556_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_5557_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5558_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5559_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5560_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5561_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5562_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5563_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_5564_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5565_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_5566_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_5567_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5568_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_5569_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_5570_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N3: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_5571_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N3: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_5572_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N3: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_5573_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_5574_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_5575_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_5576_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_5577_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
               => ( ( ord_less_real @ A @ Y4 )
                  & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_5578_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ( arctan @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_5579_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_5580_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_5581_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_5582_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
      = ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_5583_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_5584_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_5585_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_5586_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_5587_zmod__numeral__Bit0,axiom,
    ! [V: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_5588_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_5589_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_5590_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_5591_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_5592_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_5593_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_5594_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_5595_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_5596_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_5597_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q6: int] :
            ( M
            = ( times_times_int @ D @ Q6 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_5598_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q2: int] :
          ( M
          = ( times_times_int @ D @ Q2 ) ) ) ).

% zmod_eq_0D
thf(fact_5599_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_5600_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_5601_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_5602_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_5603_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_5604_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M6: int] :
          ? [N3: int] :
            ( ( ord_less_int @ M6 @ ( abs_abs_int @ N3 ) )
            & ( member_int @ N3 @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_5605_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_5606_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_5607_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_5608_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_5609_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_5610_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_5611_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_5612_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_5613_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_5614_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).

% zabs_def
thf(fact_5615_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A3 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B3 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A3 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_5616_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_5617_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_5618_div__mod__decomp__int,axiom,
    ! [A3: int,N2: int] :
      ( A3
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A3 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_5619_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_5620_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_5621_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_5622_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_5623_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ 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_5624_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ 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_5625_split__zmod,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_5626_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5627_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_5628_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_5629_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_5630_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_5631_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_5632_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_5633_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_5634_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N2 )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_5635_decr__lemma,axiom,
    ! [D: int,X: int,Z2: 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 @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).

% decr_lemma
thf(fact_5636_incr__lemma,axiom,
    ! [D: int,Z2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_5637_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N2: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_5638_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_5639_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_5640_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_5641_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_5642_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_5643_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_5644_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_5645_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_5646_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_5647_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_5648_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_5649_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_5650_sin__cos__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% sin_cos_npi
thf(fact_5651_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5652_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5653_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5654_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_5655_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_5656_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_5657_product__nth,axiom,
    ! [N2: nat,Xs2: list_int,Ys3: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs2 @ Ys3 ) @ N2 )
        = ( product_Pair_int_int @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5658_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys3 ) @ N2 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5659_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys3: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys3 ) @ N2 )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5660_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys3: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys3 ) @ N2 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5661_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys3: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys3 ) @ N2 )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5662_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys3: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys3 ) @ N2 )
        = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5663_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys3: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys3 ) @ N2 )
        = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5664_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys3: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys3 ) @ N2 )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5665_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys3: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs2 @ Ys3 ) @ N2 )
        = ( product_Pair_nat_o @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5666_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys3: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs2 @ Ys3 ) @ N2 )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).

% product_nth
thf(fact_5667_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_5668_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_5669_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_5670_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_5671_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_5672_signed__take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_5673_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_5674_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( tanh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_5675_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_5676_summable__zero,axiom,
    ( summable_real
    @ ^ [N3: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_5677_summable__zero,axiom,
    ( summable_nat
    @ ^ [N3: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_5678_summable__zero,axiom,
    ( summable_int
    @ ^ [N3: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_5679_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_5680_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_5681_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_5682_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_5683_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_5684_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_5685_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_5686_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_5687_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_5688_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_5689_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_5690_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_5691_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_5692_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_5693_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_5694_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_5695_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_5696_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_5697_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_5698_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_5699_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_5700_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_5701_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_5702_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_5703_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_5704_sin__cos__squared__add3,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_5705_sin__cos__squared__add3,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_5706_sin__npi2,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_5707_sin__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_5708_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_5709_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_5710_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_5711_signed__take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5712_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_5713_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_5714_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_5715_sin__cos__squared__add,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_5716_sin__cos__squared__add,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_5717_sin__cos__squared__add2,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_5718_sin__cos__squared__add2,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_5719_sin__2npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_5720_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
        = one_one_complex )
     => ( ( sin_complex @ X )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_5721_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
     => ( ( sin_real @ X )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_5722_sin__add,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_5723_summable__comparison__test_H,axiom,
    ! [G: nat > real,N6: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N6 @ N )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N ) ) @ ( G @ N ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_5724_summable__comparison__test_H,axiom,
    ! [G: nat > real,N6: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N6 @ N )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_5725_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N8 @ N )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_5726_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N8 @ N )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_5727_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_5728_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_5729_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_5730_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_5731_cos__add,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_5732_cos__diff,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_5733_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_5734_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_5735_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_5736_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( ( sin_complex @ X )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_5737_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_5738_powser__insidea,axiom,
    ! [F: nat > real,X: real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_5739_powser__insidea,axiom,
    ! [F: nat > complex,X: complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_5740_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_5741_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_5742_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_5743_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_5744_summable__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_5745_summable__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_5746_summable__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_5747_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_5748_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_5749_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_5750_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_5751_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_5752_cos__arctan__not__zero,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_5753_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_5754_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_5755_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_5756_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_5757_cos__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_5758_cos__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_5759_sin__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_5760_sin__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_5761_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_5762_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_5763_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_5764_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_5765_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_5766_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_5767_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_5768_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_5769_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_5770_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_5771_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_5772_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_5773_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_5774_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_5775_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_5776_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_0_powser
thf(fact_5777_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_0_powser
thf(fact_5778_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_5779_cos__mono__le__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_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_5780_cos__inj__pi,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 )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_5781_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_5782_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_5783_summable__powser__split__head,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_5784_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_5785_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_5786_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_5787_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_5788_sin__times__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_5789_sin__times__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_5790_sin__times__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_5791_sin__times__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_5792_cos__times__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_5793_cos__times__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_5794_sin__plus__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_5795_sin__plus__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_5796_sin__diff__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_5797_sin__diff__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_5798_cos__diff__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_5799_cos__diff__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_5800_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N8 @ N )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_5801_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N8 @ N )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_5802_cos__double__sin,axiom,
    ! [W2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_5803_cos__double__sin,axiom,
    ! [W2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_5804_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_5805_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_5806_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_5807_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_5808_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_5809_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_5810_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_5811_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_5812_powser__inside,axiom,
    ! [F: nat > real,X: real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_5813_powser__inside,axiom,
    ! [F: nat > complex,X: complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_5814_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_5815_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_5816_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_5817_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_5818_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_5819_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_5820_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_5821_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_5822_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_5823_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_5824_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ pi )
            & ( X
              = ( cos_real @ T3 ) )
            & ( Y
              = ( sin_real @ T3 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_5825_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_5826_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_5827_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_5828_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_5829_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_5830_cos__is__zero,axiom,
    ? [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
      & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X4 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
thf(fact_5831_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_5832_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
            & ( ord_less_eq_real @ X4 @ pi )
            & ( ( cos_real @ X4 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% cos_total
thf(fact_5833_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_5834_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T3 ) )
              & ( Y
                = ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_5835_sincos__total__2pi__le,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 )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T3 ) )
          & ( Y
            = ( sin_real @ T3 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_5836_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 )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T3 ) )
               => ( Y
                 != ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_5837_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
            @ Z2 ) ) ) ) ).

% powser_split_head(1)
thf(fact_5838_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
            @ Z2 ) ) ) ) ).

% powser_split_head(1)
thf(fact_5839_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
          @ Z2 )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_5840_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
          @ Z2 )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_5841_sin__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_5842_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N7: nat] :
            ( ( ord_less_eq_nat @ N9 @ N7 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N7 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_5843_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N7: nat] :
            ( ( ord_less_eq_nat @ N9 @ N7 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N7 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_5844_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_5845_summable__power__series,axiom,
    ! [F: nat > real,Z2: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( summable_real
              @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z2 @ I3 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_5846_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R0 @ N ) ) @ M7 )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_5847_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_5848_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_5849_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_5850_cos__times__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_5851_cos__times__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_5852_cos__plus__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_5853_cos__plus__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_5854_summable__ratio__test,axiom,
    ! [C: real,N6: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N6 @ N )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_5855_summable__ratio__test,axiom,
    ! [C: real,N6: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N6 @ N )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_5856_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_5857_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_5858_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_5859_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_5860_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_5861_cos__double__cos,axiom,
    ! [W2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_5862_cos__double__cos,axiom,
    ! [W2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_5863_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_ri631733984087533419it_int @ N2 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_5864_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_5865_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_5866_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_5867_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_5868_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_5869_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_5870_cos__ge__zero,axiom,
    ! [X: 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 @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_5871_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_5872_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_5873_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_5874_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_5875_tan__double,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_5876_tan__double,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_5877_sum__gp,axiom,
    ! [N2: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5878_sum__gp,axiom,
    ! [N2: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5879_sum__gp,axiom,
    ! [N2: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5880_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5881_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5882_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5883_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5884_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > complex,Y: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5885_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5886_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5887_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5888_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5889_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > real,Y: extended_enat > real] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5890_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5891_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5892_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5893_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5894_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > real,Y: extended_enat > real] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5895_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5896_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5897_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5898_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > rat,Y: complex > rat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5899_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: extended_enat > rat,Y: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5900_geometric__deriv__sums,axiom,
    ! [Z2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_5901_geometric__deriv__sums,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_5902_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_5903_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_5904_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_5905_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_5906_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_5907_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_5908_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_5909_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_5910_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_5911_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_5912_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_5913_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_5914_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_5915_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_5916_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_5917_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_5918_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_5919_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_5920_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_5921_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_5922_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_5923_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_5924_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_5925_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_5926_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_5927_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_5928_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_5929_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_5930_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_5931_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5932_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5933_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5934_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5935_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5936_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5937_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5938_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5939_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_5940_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_5941_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_5942_Ints__sum,axiom,
    ! [A3: set_complex,F: complex > int] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( member_int @ ( F @ X4 ) @ ring_1_Ints_int ) )
     => ( member_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ring_1_Ints_int ) ) ).

% Ints_sum
thf(fact_5943_Ints__sum,axiom,
    ! [A3: set_real,F: real > int] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( member_int @ ( F @ X4 ) @ ring_1_Ints_int ) )
     => ( member_int @ ( groups1932886352136224148al_int @ F @ A3 ) @ ring_1_Ints_int ) ) ).

% Ints_sum
thf(fact_5944_Ints__sum,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member_int @ ( F @ X4 ) @ ring_1_Ints_int ) )
     => ( member_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ring_1_Ints_int ) ) ).

% Ints_sum
thf(fact_5945_Ints__sum,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_sum
thf(fact_5946_Ints__sum,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_sum
thf(fact_5947_Ints__sum,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_sum
thf(fact_5948_Ints__sum,axiom,
    ! [A3: set_real,F: real > complex] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups5754745047067104278omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_sum
thf(fact_5949_Ints__sum,axiom,
    ! [A3: set_nat,F: nat > complex] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups2073611262835488442omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_sum
thf(fact_5950_Ints__sum,axiom,
    ! [A3: set_int,F: int > complex] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups3049146728041665814omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_sum
thf(fact_5951_Ints__sum,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups6591440286371151544t_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_sum
thf(fact_5952_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A3 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_5953_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A3 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_5954_sum_Oneutral__const,axiom,
    ! [A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A3 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_5955_sum_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A3 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_5956_of__nat__sum,axiom,
    ! [F: complex > nat,A3: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X3: complex] : ( semiri8010041392384452111omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5957_of__nat__sum,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5958_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups2073611262835488442omplex
        @ ^ [X3: nat] : ( semiri8010041392384452111omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5959_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri681578069525770553at_rat @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [X3: nat] : ( semiri681578069525770553at_rat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5960_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5961_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5962_of__nat__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X3: nat] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_sum
thf(fact_5963_sums__zero,axiom,
    ( sums_real
    @ ^ [N3: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_5964_sums__zero,axiom,
    ( sums_nat
    @ ^ [N3: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_5965_sums__zero,axiom,
    ( sums_int
    @ ^ [N3: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_5966_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5967_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5968_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5969_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5970_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5971_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5972_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5973_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5974_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5975_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5976_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5977_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5978_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5979_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5980_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5981_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5982_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5983_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5984_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_5985_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_5986_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_5987_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_5988_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_5989_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_5990_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_5991_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_5992_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_5993_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_5994_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_5995_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_5996_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_5997_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_5998_sum_Oempty,axiom,
    ! [G: extended_enat > real] :
      ( ( groups4148127829035722712t_real @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_real ) ).

% sum.empty
thf(fact_5999_sum_Oempty,axiom,
    ! [G: extended_enat > rat] :
      ( ( groups1392844769737527556at_rat @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6000_sum_Oempty,axiom,
    ! [G: extended_enat > nat] :
      ( ( groups2027974829824023292at_nat @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_6001_sum_Oempty,axiom,
    ! [G: extended_enat > int] :
      ( ( groups2025484359314973016at_int @ G @ bot_bo7653980558646680370d_enat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6002_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_6003_sum_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6004_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_6005_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6006_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6007_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6008_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6009_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6010_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 )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6011_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6012_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6013_sum_Oinfinite,axiom,
    ! [A3: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6014_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6015_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6016_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6017_sum_Oinfinite,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6018_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6019_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > rat] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6020_sum_Oinfinite,axiom,
    ! [A3: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G @ A3 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6021_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5693394587270226106ex_nat @ G @ A3 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6022_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > nat] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups2027974829824023292at_nat @ G @ A3 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6023_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_6024_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_6025_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_6026_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_6027_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_6028_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_6029_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_6030_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6031_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6032_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6033_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6034_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6035_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6036_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6037_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6038_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6039_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_6040_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6041_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6042_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6043_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6044_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6045_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6046_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6047_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6048_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6049_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_6050_sum__abs,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_6051_sum__abs,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_6052_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_6053_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_6054_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_6055_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_6056_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_6057_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_6058_tan__npi,axiom,
    ! [N2: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_6059_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_6060_sum__abs__ge__zero,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_6061_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_6062_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_6063_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_6064_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_6065_even__Suc,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% even_Suc
thf(fact_6066_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_Suc_Suc_iff
thf(fact_6067_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6068_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6069_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_6070_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_6071_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_6072_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_6073_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_6074_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6075_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6076_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6077_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6078_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_6079_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A3 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_6080_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_6081_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A3 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_6082_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_6083_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_6084_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_6085_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_6086_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_6087_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_6088_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_6089_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_6090_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_6091_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6092_even__power,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_6093_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_6094_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_6095_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_6096_power__less__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_6097_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_6098_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_6099_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_6100_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_6101_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_6102_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_6103_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_6104_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_6105_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_6106_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_6107_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_6108_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_6109_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% odd_Suc_minus_one
thf(fact_6110_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_6111_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A3 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_6112_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A3 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A3 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_6113_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A3 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A3 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_6114_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_6115_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_6116_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_6117_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_6118_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_6119_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_6120_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_6121_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_6122_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_6123_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_6124_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_6125_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6126_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6127_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6128_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6129_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6130_sum__mono,axiom,
    ! [K5: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6131_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6132_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6133_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6134_sum__mono,axiom,
    ! [K5: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6135_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6136_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6137_sum__mono,axiom,
    ! [K5: set_complex,F: complex > int,G: complex > int] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6138_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6139_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6140_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A3 ) @ ( groups3542108847815614940at_nat @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_6141_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X3: nat] : ( plus_plus_real @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A3 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A3 ) @ ( groups6591440286371151544t_real @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_6142_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X3: complex] : ( plus_plus_complex @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A3 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A3 ) @ ( groups7754918857620584856omplex @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_6143_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( plus_plus_int @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A3 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A3 ) @ ( groups4538972089207619220nt_int @ H2 @ A3 ) ) ) ).

% sum.distrib
thf(fact_6144_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ~ ( ( dvd_dvd_nat @ B @ A )
          & ( B != A ) ) ) ).

% gcd_nat.asym
thf(fact_6145_gcd__nat_Orefl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% gcd_nat.refl
thf(fact_6146_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% gcd_nat.trans
thf(fact_6147_gcd__nat_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( dvd_dvd_nat @ A2 @ B2 )
          & ( dvd_dvd_nat @ B2 @ A2 ) ) ) ) ).

% gcd_nat.eq_iff
thf(fact_6148_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ A @ A )
        & ( A != A ) ) ).

% gcd_nat.irrefl
thf(fact_6149_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( A = B ) ) ) ).

% gcd_nat.antisym
thf(fact_6150_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( ( ( dvd_dvd_nat @ B @ C )
          & ( B != C ) )
       => ( ( dvd_dvd_nat @ A @ C )
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
thf(fact_6151_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( ( dvd_dvd_nat @ B @ C )
          & ( B != C ) )
       => ( ( dvd_dvd_nat @ A @ C )
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
thf(fact_6152_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( ( dvd_dvd_nat @ A @ C )
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
thf(fact_6153_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% gcd_nat.strict_iff_not
thf(fact_6154_gcd__nat_Oorder__iff__strict,axiom,
    ( dvd_dvd_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ( dvd_dvd_nat @ A2 @ B2 )
            & ( A2 != B2 ) )
          | ( A2 = B2 ) ) ) ) ).

% gcd_nat.order_iff_strict
thf(fact_6155_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
thf(fact_6156_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( dvd_dvd_nat @ A @ B ) ) ).

% gcd_nat.strict_implies_order
thf(fact_6157_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
thf(fact_6158_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( dvd_dvd_nat @ A @ B )
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
thf(fact_6159_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_6160_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_6161_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6162_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_6163_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_6164_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A3: set_complex] :
      ( ( ( groups5808333547571424918x_real @ G @ A3 )
       != zero_zero_real )
     => ~ ! [A4: complex] :
            ( ( member_complex @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6165_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A3: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A3 )
       != zero_zero_real )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6166_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A3: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A3 )
       != zero_zero_real )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6167_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > rat,A3: set_complex] :
      ( ( ( groups5058264527183730370ex_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: complex] :
            ( ( member_complex @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6168_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A3: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6169_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A3: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6170_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A3: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G @ A3 )
       != zero_zero_rat )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6171_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > nat,A3: set_complex] :
      ( ( ( groups5693394587270226106ex_nat @ G @ A3 )
       != zero_zero_nat )
     => ~ ! [A4: complex] :
            ( ( member_complex @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6172_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > nat,A3: set_real] :
      ( ( ( groups1935376822645274424al_nat @ G @ A3 )
       != zero_zero_nat )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6173_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > nat,A3: set_int] :
      ( ( ( groups4541462559716669496nt_nat @ G @ A3 )
       != zero_zero_nat )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6174_sum_Oneutral,axiom,
    ! [A3: set_nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ( G @ X4 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A3 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_6175_sum_Oneutral,axiom,
    ! [A3: set_nat,G: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ( G @ X4 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A3 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_6176_sum_Oneutral,axiom,
    ! [A3: set_complex,G: complex > complex] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ( G @ X4 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A3 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_6177_sum_Oneutral,axiom,
    ! [A3: set_int,G: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ( G @ X4 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A3 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_6178_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6179_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6180_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6181_sums__0,axiom,
    ! [F: nat > real] :
      ( ! [N: nat] :
          ( ( F @ N )
          = zero_zero_real )
     => ( sums_real @ F @ zero_zero_real ) ) ).

% sums_0
thf(fact_6182_sums__0,axiom,
    ! [F: nat > nat] :
      ( ! [N: nat] :
          ( ( F @ N )
          = zero_zero_nat )
     => ( sums_nat @ F @ zero_zero_nat ) ) ).

% sums_0
thf(fact_6183_sums__0,axiom,
    ! [F: nat > int] :
      ( ! [N: nat] :
          ( ( F @ N )
          = zero_zero_int )
     => ( sums_int @ F @ zero_zero_int ) ) ).

% sums_0
thf(fact_6184_sums__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6185_sums__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6186_sums__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6187_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6188_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6189_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_6190_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).

% sums_If_finite_set
thf(fact_6191_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).

% sums_If_finite_set
thf(fact_6192_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).

% sums_If_finite_set
thf(fact_6193_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_6194_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_6195_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_6196_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_6197_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A2: real,B2: real] :
          ( ( A2 = zero_zero_real )
         => ( B2 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_6198_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( A2 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_6199_dvd__productE,axiom,
    ! [P6: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P6
              = ( times_times_nat @ X4 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X4 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_6200_dvd__productE,axiom,
    ! [P6: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
     => ~ ! [X4: int,Y3: int] :
            ( ( P6
              = ( times_times_int @ X4 @ Y3 ) )
           => ( ( dvd_dvd_int @ X4 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_6201_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B8: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B8 @ C5 ) )
          & ( dvd_dvd_nat @ B8 @ B )
          & ( dvd_dvd_nat @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_6202_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B8: int,C5: int] :
          ( ( A
            = ( times_times_int @ B8 @ C5 ) )
          & ( dvd_dvd_int @ B8 @ B )
          & ( dvd_dvd_int @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_6203_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B @ K2 ) ) ) ).

% dvdE
thf(fact_6204_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_6205_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_6206_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_6207_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_6208_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_6209_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_6210_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_6211_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B2: real,A2: real] :
        ? [K3: real] :
          ( A2
          = ( times_times_real @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6212_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B2: rat,A2: rat] :
        ? [K3: rat] :
          ( A2
          = ( times_times_rat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6213_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A2: nat] :
        ? [K3: nat] :
          ( A2
          = ( times_times_nat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6214_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A2: int] :
        ? [K3: int] :
          ( A2
          = ( times_times_int @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6215_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6216_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6217_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6218_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_6219_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6220_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6221_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6222_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_6223_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6224_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6225_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6226_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_6227_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6228_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6229_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6230_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_6231_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6232_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6233_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6234_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_6235_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6236_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6237_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6238_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_6239_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6240_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6241_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6242_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_6243_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6244_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6245_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6246_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_6247_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6248_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6249_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6250_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6251_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6252_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6253_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6254_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_6255_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_6256_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_6257_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_6258_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_6259_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_6260_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_6261_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_6262_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_6263_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_6264_dvd__diff,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z2 )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_6265_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z2 )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_6266_dvd__diff,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z2 )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_6267_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6268_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6269_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6270_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6271_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6272_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6273_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6274_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6275_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_6276_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_6277_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_6278_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_6279_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_6280_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_6281_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_6282_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6283_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6284_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( ( abs_abs_rat @ L )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6285_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6286_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_6287_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_6288_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_6289_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6290_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6291_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6292_sum__nonpos,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6293_sum__nonpos,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6294_sum__nonpos,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6295_sum__nonpos,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6296_sum__nonpos,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6297_sum__nonpos,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6298_sum__nonpos,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6299_sum__nonpos,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6300_sum__nonpos,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6301_sum__nonpos,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6302_sum__nonneg,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6303_sum__nonneg,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6304_sum__nonneg,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6305_sum__nonneg,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6306_sum__nonneg,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6307_sum__nonneg,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6308_sum__nonneg,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6309_sum__nonneg,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6310_sum__nonneg,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6311_sum__nonneg,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_6312_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_6313_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6314_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6315_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6316_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6317_sum__mono__inv,axiom,
    ! [F: extended_enat > rat,I5: set_Extended_enat,G: extended_enat > rat,I: extended_enat] :
      ( ( ( groups1392844769737527556at_rat @ F @ I5 )
        = ( groups1392844769737527556at_rat @ G @ I5 ) )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_Extended_enat @ I @ I5 )
         => ( ( finite4001608067531595151d_enat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6318_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6319_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6320_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6321_sum__mono__inv,axiom,
    ! [F: extended_enat > nat,I5: set_Extended_enat,G: extended_enat > nat,I: extended_enat] :
      ( ( ( groups2027974829824023292at_nat @ F @ I5 )
        = ( groups2027974829824023292at_nat @ G @ I5 ) )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_Extended_enat @ I @ I5 )
         => ( ( finite4001608067531595151d_enat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6322_sum__mono__inv,axiom,
    ! [F: real > int,I5: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I5 )
        = ( groups1932886352136224148al_int @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6323_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ ( suc @ X4 ) @ A3 )
           => ( ( F @ ( suc @ X4 ) )
              = ( G @ ( suc @ X4 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A3 )
          = ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ).

% sum_cong_Suc
thf(fact_6324_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ ( suc @ X4 ) @ A3 )
           => ( ( F @ ( suc @ X4 ) )
              = ( G @ ( suc @ X4 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A3 )
          = ( groups6591440286371151544t_real @ G @ A3 ) ) ) ) ).

% sum_cong_Suc
thf(fact_6325_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6326_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6327_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6328_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6329_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6330_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6331_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6332_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6333_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6334_sum_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > real,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups4148127829035722712t_real
          @ ^ [X3: extended_enat] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6335_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6336_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6337_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6338_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X3: complex] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6339_sum_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > rat,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups1392844769737527556at_rat
          @ ^ [X3: extended_enat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6340_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: real > nat,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X3: real] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_6341_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A3: set_nat,S4: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( S4
            = ( plus_plus_real @ S3
              @ ( groups6591440286371151544t_real
                @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
                @ A3 ) ) )
         => ( sums_real
            @ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A3 ) @ ( F @ N3 ) @ ( G @ N3 ) )
            @ S4 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_6342_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6343_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6344_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6345_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( ( dvd_dvd_real @ A @ B )
        & ~ ( dvd_dvd_real @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6346_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6347_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6348_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X3: real] :
                ( ( member_real @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6349_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6350_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6351_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6352_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X3: real] :
                ( ( member_real @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6353_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6354_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6355_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6356_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups1392844769737527556at_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6357_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A3 )
            = zero_zero_nat )
          = ( ! [X3: real] :
                ( ( member_real @ X3 @ A3 )
               => ( ( F @ X3 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6358_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: 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 )
                    & ( ( I @ 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_6359_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: 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 )
                    & ( ( I @ 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_6360_sum__le__included,axiom,
    ! [S: set_int,T: set_Extended_enat,G: extended_enat > real,I: extended_enat > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups4148127829035722712t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6361_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: 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 )
                    & ( ( I @ 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_6362_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: 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 )
                    & ( ( I @ 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_6363_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G: extended_enat > real,I: extended_enat > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups4148127829035722712t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6364_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_int,G: int > real,I: int > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6365_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_complex,G: complex > real,I: complex > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6366_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_Extended_enat,G: extended_enat > real,I: extended_enat > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S ) @ ( groups4148127829035722712t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6367_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: 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 )
                    & ( ( I @ 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_6368_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6369_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A3 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6370_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6371_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6372_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6373_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6374_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6375_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A3 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6376_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A3 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6377_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X4: extended_enat] :
            ( ( member_Extended_enat @ X4 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A3 )
              & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6378_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6379_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6380_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups4148127829035722712t_real @ H2 @ S3 ) @ ( groups4148127829035722712t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6381_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6382_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6383_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6384_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_Extended_enat,H2: extended_enat > rat,G: extended_enat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups1392844769737527556at_rat @ H2 @ S3 ) @ ( groups1392844769737527556at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6385_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6386_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6387_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups2027974829824023292at_nat @ H2 @ S3 ) @ ( groups2027974829824023292at_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6388_sum__strict__mono,axiom,
    ! [A3: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6389_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6390_sum__strict__mono,axiom,
    ! [A3: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6391_sum__strict__mono,axiom,
    ! [A3: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6392_sum__strict__mono,axiom,
    ! [A3: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6393_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6394_sum__strict__mono,axiom,
    ! [A3: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6395_sum__strict__mono,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6396_sum__strict__mono,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A3 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6397_sum__strict__mono,axiom,
    ! [A3: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ G @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6398_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S3: set_real,I: real > real,J: real > real,T5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6399_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S3: set_real,I: int > real,J: real > int,T5: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6400_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S3: set_real,I: complex > real,J: real > complex,T5: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6401_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_Extended_enat,S3: set_real,I: extended_enat > real,J: real > extended_enat,T5: set_Extended_enat,G: real > real,H2: extended_enat > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite4001608067531595151d_enat @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T5 @ T4 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups4148127829035722712t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6402_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S3: set_int,I: real > int,J: int > real,T5: set_real,G: int > real,H2: real > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6403_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S3: set_int,I: int > int,J: int > int,T5: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6404_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,S3: set_int,I: complex > int,J: int > complex,T5: set_complex,G: int > real,H2: complex > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6405_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_Extended_enat,S3: set_int,I: extended_enat > int,J: int > extended_enat,T5: set_Extended_enat,G: int > real,H2: extended_enat > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite4001608067531595151d_enat @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T5 @ T4 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups4148127829035722712t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6406_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_real,S3: set_complex,I: real > complex,J: complex > real,T5: set_real,G: complex > real,H2: real > real] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5808333547571424918x_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6407_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_int,S3: set_complex,I: int > complex,J: complex > int,T5: set_int,G: complex > real,H2: int > real] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5808333547571424918x_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6408_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_6409_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_6410_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_6411_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_6412_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_6413_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_6414_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_6415_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_6416_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_6417_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z @ X2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_6418_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_6419_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_6420_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_6421_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_6422_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_6423_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_6424_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_6425_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_6426_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_6427_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_6428_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_6429_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_6430_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_6431_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_6432_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_6433_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6434_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6435_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6436_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6437_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6438_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6439_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6440_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6441_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6442_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6443_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6444_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6445_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6446_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6447_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6448_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6449_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6450_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6451_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6452_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6453_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6454_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6455_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6456_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6457_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6458_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6459_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6460_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6461_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_6462_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_6463_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_6464_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_6465_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_6466_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_6467_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_6468_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_6469_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_6470_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_6471_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_6472_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_6473_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_6474_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_6475_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_6476_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_6477_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6478_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6479_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_6480_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_6481_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_6482_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A2 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6483_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A2: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A2 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6484_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6485_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_6486_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_6487_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6488_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6489_dvd__power__le,axiom,
    ! [X: real,Y: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6490_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6491_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6492_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6493_power__le__dvd,axiom,
    ! [A: real,N2: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6494_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6495_power__le__dvd,axiom,
    ! [A: complex,N2: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_6496_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6497_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6498_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6499_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_6500_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_6501_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_6502_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_6503_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_6504_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_6505_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_6506_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B3: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6507_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B3: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6508_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B3: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6509_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real,B3: real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S )
            = B3 )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6510_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B3: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6511_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B3: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B3 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6512_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B3: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6513_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B3: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6514_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > rat,B3: rat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1392844769737527556at_rat @ F @ S )
            = B3 )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6515_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B3: nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6516_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6517_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6518_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6519_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6520_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6521_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6522_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6523_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6524_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > rat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1392844769737527556at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6525_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6526_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_6527_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_6528_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_6529_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_6530_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( times_times_nat @ A @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) )
        | ( ( times_times_nat @ B @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_nat
thf(fact_6531_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y3 ) )
          = D5 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y3 ) )
          = D5 ) ) ) ).

% bezout1_nat
thf(fact_6532_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X3: real] :
                  ( ( G @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6533_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6534_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A3
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6535_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G
          @ ( minus_925952699566721837d_enat @ A3
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups4148127829035722712t_real @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6536_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X3: real] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups1300246762558778688al_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6537_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6538_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A3
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6539_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G
          @ ( minus_925952699566721837d_enat @ A3
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups1392844769737527556at_rat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6540_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: real > nat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X3: real] :
                  ( ( G @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6541_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6542_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6543_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6544_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6545_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6546_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6547_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6548_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6549_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6550_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6551_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6552_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6553_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6554_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6555_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6556_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6557_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6558_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6559_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6560_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6561_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6562_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6563_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6564_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6565_sum__pos,axiom,
    ! [I5: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6566_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T5 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6567_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T5 )
              = ( groups8778361861064173332t_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6568_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T5 )
              = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6569_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4148127829035722712t_real @ G @ T5 )
              = ( groups4148127829035722712t_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6570_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T5 )
              = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6571_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T5 )
              = ( groups3906332499630173760nt_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6572_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T5 )
              = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6573_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > rat,H2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1392844769737527556at_rat @ G @ T5 )
              = ( groups1392844769737527556at_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6574_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T5 )
              = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6575_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ T5 )
              = ( groups4541462559716669496nt_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6576_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6577_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S3 )
              = ( groups8778361861064173332t_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6578_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6579_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4148127829035722712t_real @ G @ S3 )
              = ( groups4148127829035722712t_real @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6580_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6581_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S3 )
              = ( groups3906332499630173760nt_rat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6582_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
              = ( groups5058264527183730370ex_rat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6583_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > rat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1392844769737527556at_rat @ G @ S3 )
              = ( groups1392844769737527556at_rat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6584_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S3 )
              = ( groups1935376822645274424al_nat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6585_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ S3 )
              = ( groups4541462559716669496nt_nat @ H2 @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6586_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T5 )
            = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6587_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T5 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6588_sum_Omono__neutral__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G @ T5 )
            = ( groups4148127829035722712t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6589_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T5 )
            = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6590_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T5 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6591_sum_Omono__neutral__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups1392844769737527556at_rat @ G @ T5 )
            = ( groups1392844769737527556at_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6592_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T5 )
            = ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6593_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T5 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6594_sum_Omono__neutral__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G @ T5 )
            = ( groups2027974829824023292at_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6595_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T5 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6596_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S3 )
            = ( groups8778361861064173332t_real @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6597_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6598_sum_Omono__neutral__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G @ S3 )
            = ( groups4148127829035722712t_real @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6599_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S3 )
            = ( groups3906332499630173760nt_rat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6600_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6601_sum_Omono__neutral__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups1392844769737527556at_rat @ G @ S3 )
            = ( groups1392844769737527556at_rat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6602_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S3 )
            = ( groups4541462559716669496nt_nat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6603_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6604_sum_Omono__neutral__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G @ S3 )
            = ( groups2027974829824023292at_nat @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6605_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6606_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A3 )
                  = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6607_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) )
               => ( ( groups8778361861064173332t_real @ G @ A3 )
                  = ( groups8778361861064173332t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6608_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A3 )
                  = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6609_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G @ C4 )
                  = ( groups4148127829035722712t_real @ H2 @ C4 ) )
               => ( ( groups4148127829035722712t_real @ G @ A3 )
                  = ( groups4148127829035722712t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6610_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A3 )
                  = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6611_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A3 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6612_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A3 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6613_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > rat,H2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups1392844769737527556at_rat @ G @ C4 )
                  = ( groups1392844769737527556at_rat @ H2 @ C4 ) )
               => ( ( groups1392844769737527556at_rat @ G @ A3 )
                  = ( groups1392844769737527556at_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6614_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A3 )
                  = ( groups1935376822645274424al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6615_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) )
               => ( ( groups4541462559716669496nt_nat @ G @ A3 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6616_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A3 )
                  = ( groups8097168146408367636l_real @ H2 @ B3 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6617_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A3 )
                  = ( groups8778361861064173332t_real @ H2 @ B3 ) )
                = ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6618_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A3 )
                  = ( groups5808333547571424918x_real @ H2 @ B3 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6619_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G @ A3 )
                  = ( groups4148127829035722712t_real @ H2 @ B3 ) )
                = ( ( groups4148127829035722712t_real @ G @ C4 )
                  = ( groups4148127829035722712t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6620_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A3 )
                  = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6621_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A3 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B3 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6622_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A3 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B3 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6623_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > rat,H2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups1392844769737527556at_rat @ G @ A3 )
                  = ( groups1392844769737527556at_rat @ H2 @ B3 ) )
                = ( ( groups1392844769737527556at_rat @ G @ C4 )
                  = ( groups1392844769737527556at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6624_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A3 )
                  = ( groups1935376822645274424al_nat @ H2 @ B3 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6625_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ A3 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B3 ) )
                = ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6626_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups8778361861064173332t_real @ G @ A3 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6627_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A3: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B3 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5808333547571424918x_real @ G @ A3 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A3 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6628_sum_Osubset__diff,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,G: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B3 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups4148127829035722712t_real @ G @ A3 )
          = ( plus_plus_real @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A3 @ B3 ) ) @ ( groups4148127829035722712t_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6629_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups3906332499630173760nt_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6630_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A3: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5058264527183730370ex_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A3 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6631_sum_Osubset__diff,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,G: extended_enat > rat] :
      ( ( ord_le7203529160286727270d_enat @ B3 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups1392844769737527556at_rat @ G @ A3 )
          = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G @ ( minus_925952699566721837d_enat @ A3 @ B3 ) ) @ ( groups1392844769737527556at_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6632_sum_Osubset__diff,axiom,
    ! [B3: set_int,A3: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B3 @ A3 )
     => ( ( finite_finite_int @ A3 )
       => ( ( groups4541462559716669496nt_nat @ G @ A3 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A3 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6633_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A3: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5693394587270226106ex_nat @ G @ A3 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A3 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6634_sum_Osubset__diff,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,G: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B3 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups2027974829824023292at_nat @ G @ A3 )
          = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A3 @ B3 ) ) @ ( groups2027974829824023292at_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6635_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A3: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B3 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5690904116761175830ex_int @ G @ A3 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A3 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6636_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6637_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6638_sums__Suc__imp,axiom,
    ! [F: nat > real,S: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
          @ S )
       => ( sums_real @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_6639_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6640_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6641_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6642_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_6643_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ( F @ I2 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_6644_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X3: real] : ( P @ ( times_times_real @ L @ X3 ) ) )
      = ( ? [X3: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X3 @ zero_zero_real ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6645_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
      = ( ? [X3: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6646_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6647_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
      = ( ? [X3: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6648_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_6649_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_6650_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6651_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6652_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6653_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6654_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6655_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6656_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6657_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6658_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_6659_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_6660_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X2: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6661_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X2: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6662_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6663_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X2: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6664_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X2: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6665_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6666_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6667_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6668_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6669_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6670_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_6671_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_6672_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6673_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6674_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6675_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6676_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_6677_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_6678_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6679_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6680_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_6681_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_6682_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_6683_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_6684_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_6685_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_6686_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X4: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D5 @ A )
          & ( dvd_dvd_nat @ D5 @ B )
          & ( ( times_times_nat @ A @ X4 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_6687_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_6688_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q4 )
          = ( modulo_modulo_nat @ N2 @ Q4 ) )
        = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_6689_real__of__nat__div,axiom,
    ! [D: nat,N2: nat] :
      ( ( dvd_dvd_nat @ D @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_6690_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6691_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6692_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6693_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6694_sum__mono2,axiom,
    ! [B3: set_real,A3: set_real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6695_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > real] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6696_sum__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6697_sum__mono2,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B3 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B3 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B3 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6698_sum__mono2,axiom,
    ! [B3: set_real,A3: set_real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6699_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6700_sum__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6701_sum__mono2,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ B3 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B3 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6702_sum__mono2,axiom,
    ! [B3: set_real,A3: set_real,F: real > nat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6703_sum__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6704_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_6705_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6706_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6707_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_6708_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6709_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6710_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N6 )
           => ( ( F @ N )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_6711_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D6: nat] : ( dvd_dvd_nat @ D6 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_6712_powser__sums__if,axiom,
    ! [M: nat,Z2: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N3 ) )
      @ ( power_power_complex @ Z2 @ M ) ) ).

% powser_sums_if
thf(fact_6713_powser__sums__if,axiom,
    ! [M: nat,Z2: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N3 ) )
      @ ( power_power_real @ Z2 @ M ) ) ).

% powser_sums_if
thf(fact_6714_powser__sums__if,axiom,
    ! [M: nat,Z2: int] :
      ( sums_int
      @ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N3 ) )
      @ ( power_power_int @ Z2 @ M ) ) ).

% powser_sums_if
thf(fact_6715_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6716_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6717_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6718_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6719_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6720_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6721_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6722_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6723_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6724_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6725_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6726_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6727_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6728_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6729_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6730_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6731_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6732_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6733_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_6734_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_6735_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_6736_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_6737_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_6738_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_6739_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B4 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B4 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6740_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( ( dvd_dvd_int @ B4 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B4 )
                 => ( ( ( divide_divide_int @ one_one_int @ B4 )
                      = A )
                   => ( ( ( times_times_int @ A @ B4 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6741_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_6742_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_6743_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_6744_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_6745_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_6746_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_6747_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_6748_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_6749_dvd__power,axiom,
    ! [N2: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_6750_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_6751_dvd__power,axiom,
    ! [N2: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_6752_sum__strict__mono2,axiom,
    ! [B3: set_real,A3: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6753_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X4: int] :
                  ( ( member_int @ X4 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6754_sum__strict__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X4: complex] :
                  ( ( member_complex @ X4 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6755_sum__strict__mono2,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,B: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B3 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B3 )
       => ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B3 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X4: extended_enat] :
                  ( ( member_Extended_enat @ X4 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
             => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6756_sum__strict__mono2,axiom,
    ! [B3: set_real,A3: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6757_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: int] :
                  ( ( member_int @ X4 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6758_sum__strict__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: complex] :
                  ( ( member_complex @ X4 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6759_sum__strict__mono2,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,B: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ B3 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B3 )
       => ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B3 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: extended_enat] :
                  ( ( member_Extended_enat @ X4 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6760_sum__strict__mono2,axiom,
    ! [B3: set_real,A3: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A3 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6761_sum__strict__mono2,axiom,
    ! [B3: set_int,A3: set_int,B: int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A3 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X4: int] :
                  ( ( member_int @ X4 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6762_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_6763_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_6764_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_6765_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_6766_dvd__minus__add,axiom,
    ! [Q4: nat,N2: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q4 @ N2 )
     => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q4 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_6767_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_6768_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N2 )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_6769_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6770_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6771_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6772_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6773_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6774_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6775_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6776_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6777_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6778_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6779_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6780_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X @ I5 )
          = one_one_real )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6781_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ I5 )
          = one_one_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6782_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ I5 )
          = one_one_real )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6783_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I3: complex] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6784_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6785_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6786_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6787_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6788_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6789_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6790_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_6791_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_6792_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6793_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6794_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6795_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6796_power__mono__odd,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_6797_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_6798_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_6799_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N: nat] :
              ( ( member_nat @ N @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_6800_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N: nat] :
              ( ( member_nat @ N @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_6801_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N: nat] :
              ( ( member_nat @ N @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_6802_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% odd_pos
thf(fact_6803_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_6804_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6805_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6806_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_6807_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6808_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6809_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6810_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6811_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_6812_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_6813_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_6814_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_6815_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_6816_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_6817_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_6818_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_6819_zero__le__even__power,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_6820_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_6821_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_6822_zero__le__odd__power,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6823_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6824_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6825_zero__le__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6826_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6827_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6828_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_6829_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_6830_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_6831_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_6832_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_6833_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6834_power__mono__even,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6835_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6836_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6837_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_6838_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_6839_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_6840_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6841_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6842_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6843_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E2: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M2 )
                 => ! [N7: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_6844_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E2: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M2 )
                 => ! [N7: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_6845_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_6846_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_6847_power__half__series,axiom,
    ( sums_real
    @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_6848_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).

% even_set_encode_iff
thf(fact_6849_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_6850_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_6851_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_6852_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_6853_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_6854_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_6855_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 )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
thf(fact_6856_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X3: nat] :
          ( collect_nat
          @ ^ [N3: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_6857_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6858_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6859_zero__less__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6860_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6861_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6862_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6863_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6864_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6865_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6866_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6867_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6868_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6869_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6870_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_6871_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_6872_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_6873_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_6874_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_6875_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_6876_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_6877_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_6878_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_6879_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_6880_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_6881_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_6882_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_6883_power__le__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6884_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6885_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6886_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_6887_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_6888_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_6889_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_6890_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_6891_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_6892_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_6893_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_6894_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_6895_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z )
          & ( ord_less_real @ Z @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_6896_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_6897_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_6898_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum
thf(fact_6899_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_6900_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_6901_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_6902_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_6903_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_6904_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6905_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6906_double__arith__series,axiom,
    ! [A: rat,D: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6907_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6908_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6909_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6910_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_6911_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_6912_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_6913_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6914_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6915_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6916_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6917_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6918_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6919_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6920_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6921_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6922_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6923_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6924_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6925_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6926_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6927_sum__gp__offset,axiom,
    ! [X: real,M: nat,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6928_tan__half,axiom,
    ( tan_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_6929_tan__half,axiom,
    ( tan_complex
    = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_6930_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6931_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6932_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_6933_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_6934_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_6935_lemma__termdiff2,axiom,
    ! [H2: complex,Z2: complex,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q6: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q6 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6936_lemma__termdiff2,axiom,
    ! [H2: rat,Z2: rat,N2: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N2 ) @ ( power_power_rat @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q6: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q6 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6937_lemma__termdiff2,axiom,
    ! [H2: real,Z2: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q6: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q6 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6938_diffs__equiv,axiom,
    ! [C: nat > complex,X: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_6939_diffs__equiv,axiom,
    ! [C: nat > real,X: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_6940_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 ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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.elims
thf(fact_6941_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_6942_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_6943_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_6944_intind,axiom,
    ! [I: nat,N2: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_6945_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_6946_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_6947_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_6948_lessThan__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( set_or5984915006950818249n_real @ X )
        = ( set_or5984915006950818249n_real @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_6949_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6950_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6951_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_6952_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_6953_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6954_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_6955_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_6956_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_6957_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_6958_int__dvd__int__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% int_dvd_int_iff
thf(fact_6959_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X )
        = ( replicate_VEBT_VEBT @ N2 @ Y ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_6960_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_6961_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_6962_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(80)
thf(fact_6963_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_6964_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_6965_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_6966_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_6967_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_6968_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_6969_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_6970_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_6971_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_6972_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_6973_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_6974_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_6975_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_6976_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_6977_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_6978_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_6979_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_6980_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_6981_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_6982_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_6983_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_6984_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
           => ( P @ X3 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_6985_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P @ X3 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_6986_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P @ X3 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_6987_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
            & ( P @ X3 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_6988_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ? [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P @ X3 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_6989_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P @ X3 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_6990_in__set__replicate,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6991_in__set__replicate,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6992_in__set__replicate,axiom,
    ! [X: set_nat,N2: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6993_in__set__replicate,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6994_in__set__replicate,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6995_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6996_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_6997_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_6998_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(81)
thf(fact_6999_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_7000_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_7001_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7002_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7003_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7004_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_7005_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7006_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7007_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7008_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_7009_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_Suc
thf(fact_7010_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_7011_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_7012_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_7013_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_7014_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7015_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7016_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7017_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7018_zdiv__numeral__Bit1,axiom,
    ! [V: num,W2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_7019_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(79)
thf(fact_7020_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(74)
thf(fact_7021_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7022_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7023_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7024_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7025_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_7026_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_7027_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_7028_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_7029_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7030_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7031_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7032_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7033_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7034_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_7035_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_7036_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_7037_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_7038_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_7039_int__sum,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% int_sum
thf(fact_7040_int__sum,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% int_sum
thf(fact_7041_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_7042_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N2: real] :
      ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X3: real] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
          @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7043_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7044_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_7045_lessThan__non__empty,axiom,
    ! [X: real] :
      ( ( set_or5984915006950818249n_real @ X )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_7046_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_7047_infinite__Iio,axiom,
    ! [A: real] :
      ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).

% infinite_Iio
thf(fact_7048_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ N2 )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_7049_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ N2 )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_7050_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ N2 )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_7051_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ N2 )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_7052_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ N2 )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_7053_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_7054_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_7055_uminus__dvd__conv_I2_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D6: int,T2: int] : ( dvd_dvd_int @ D6 @ ( uminus_uminus_int @ T2 ) ) ) ) ).

% uminus_dvd_conv(2)
thf(fact_7056_uminus__dvd__conv_I1_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D6: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D6 ) ) ) ) ).

% uminus_dvd_conv(1)
thf(fact_7057_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_7058_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_7059_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7060_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7061_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7062_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7063_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7064_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7065_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_7066_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_7067_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_7068_Iio__eq__empty__iff,axiom,
    ! [N2: extended_enat] :
      ( ( ( set_or8419480210114673929d_enat @ N2 )
        = bot_bo7653980558646680370d_enat )
      = ( N2 = bot_bo4199563552545308370d_enat ) ) ).

% Iio_eq_empty_iff
thf(fact_7069_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_7070_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_7071_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_7072_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_7073_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_7074_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_7075_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_7076_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_7077_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_7078_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_7079_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7080_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7081_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7082_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_7083_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_7084_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_7085_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_7086_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_7087_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
      = ( ord_less_rat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7088_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7089_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7090_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7091_lessThan__strict__subset__iff,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
      = ( ord_less_real @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7092_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_mono
thf(fact_7093_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_mono
thf(fact_7094_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_mono
thf(fact_7095_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono
thf(fact_7096_replicate__length__same,axiom,
    ! [Xs2: list_int,X: int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
         => ( X4 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_7097_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X4 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_7098_replicate__length__same,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
         => ( X4 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_7099_replicate__length__same,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( X4 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_7100_replicate__eqI,axiom,
    ! [Xs2: list_complex,N2: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N2 )
     => ( ! [Y3: complex] :
            ( ( member_complex @ Y3 @ ( set_complex2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_complex @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7101_replicate__eqI,axiom,
    ! [Xs2: list_real,N2: nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N2 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_real @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7102_replicate__eqI,axiom,
    ! [Xs2: list_set_nat,N2: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs2 )
        = N2 )
     => ( ! [Y3: set_nat] :
            ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_set_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7103_replicate__eqI,axiom,
    ! [Xs2: list_int,N2: nat,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N2 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_int @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7104_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7105_replicate__eqI,axiom,
    ! [Xs2: list_o,N2: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N2 )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_o @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7106_replicate__eqI,axiom,
    ! [Xs2: list_nat,N2: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N2 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7107_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_7108_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_7109_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_7110_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ( dvd_dvd_int @ M @ N2 )
         => ( ( dvd_dvd_int @ N2 @ M )
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_7111_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N2 )
       => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).

% zdvd_not_zless
thf(fact_7112_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_7113_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_7114_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_7115_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_7116_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: 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 @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_7117_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_7118_sum__subtractf__nat,axiom,
    ! [A3: set_complex,G: complex > nat,F: complex > nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X3: complex] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7119_sum__subtractf__nat,axiom,
    ! [A3: set_real,G: real > nat,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7120_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X3: set_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( groups8294997508430121362at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7121_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: int > nat,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X3: int] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7122_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7123_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D6: int] : ( dvd_dvd_int @ D6 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_7124_sum__eq__Suc0__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A3 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7125_sum__eq__Suc0__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7126_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( groups977919841031483927at_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A3 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7127_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A3 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: extended_enat] :
                  ( ( member_Extended_enat @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7128_sum__eq__Suc0__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7129_sum__SucD,axiom,
    ! [F: nat > nat,A3: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A3 )
        = ( suc @ N2 ) )
     => ? [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).

% sum_SucD
thf(fact_7130_sum__eq__1__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A3 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7131_sum__eq__1__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7132_sum__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( groups977919841031483927at_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A3 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7133_sum__eq__1__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A3 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y2: extended_enat] :
                  ( ( member_Extended_enat @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7134_sum__eq__1__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A3 )
                 => ( ( X3 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7135_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_7136_sqrt__add__le__add__sqrt,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 @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_7137_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > literal > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_literal ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_literal ) ) ) ) ).

% Maclaurin_zero
thf(fact_7138_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_7139_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > rat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7140_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7141_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_7142_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B9: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
              @ ( set_ord_lessThan_nat @ N2 ) )
            @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_7143_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_7144_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_Bit1
thf(fact_7145_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_7146_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_7147_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_7148_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_7149_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_7150_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7151_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7152_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7153_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_7154_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_7155_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_7156_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_7157_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_7158_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_7159_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_7160_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_7161_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_7162_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_7163_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_7164_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_7165_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_7166_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7167_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7168_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7169_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_7170_zdvd__imp__le,axiom,
    ! [Z2: int,N2: int] :
      ( ( dvd_dvd_int @ Z2 @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z2 @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_7171_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_7172_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_7173_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_7174_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_7175_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_code(3)
thf(fact_7176_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_7177_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_7178_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_7179_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_7180_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_7181_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_7182_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_7183_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_7184_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_7185_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_7186_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_7187_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_7188_suminf__le__const,axiom,
    ! [F: nat > int,X: int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7189_suminf__le__const,axiom,
    ! [F: nat > nat,X: nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7190_suminf__le__const,axiom,
    ! [F: nat > real,X: real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7191_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C2: nat > complex,N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_7192_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C2: nat > real,N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_7193_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C2: nat > rat,N3: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_7194_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C2: nat > int,N3: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_7195_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7196_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7197_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7198_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7199_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7200_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7201_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7202_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7203_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7204_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7205_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
        = ( ( abs_abs_int @ N2 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7206_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X: int] :
      ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7207_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7208_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7209_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_7210_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_7211_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_7212_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_7213_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_7214_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N2: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_7215_bset_I9_J,axiom,
    ! [D: int,D4: int,B3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_7216_bset_I10_J,axiom,
    ! [D: int,D4: int,B3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_7217_aset_I9_J,axiom,
    ! [D: int,D4: int,A3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_7218_aset_I10_J,axiom,
    ! [D: int,D4: int,A3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_7219_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_7220_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_7221_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_7222_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_7223_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_7224_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_7225_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_7226_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_7227_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_7228_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_7229_sum__count__set,axiom,
    ! [Xs2: list_int,X6: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ X6 )
     => ( ( finite_finite_int @ X6 )
       => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs2 ) @ X6 )
          = ( size_size_list_int @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7230_sum__count__set,axiom,
    ! [Xs2: list_complex,X6: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ X6 )
     => ( ( finite3207457112153483333omplex @ X6 )
       => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs2 ) @ X6 )
          = ( size_s3451745648224563538omplex @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7231_sum__count__set,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ X6 )
     => ( ( finite6177210948735845034at_nat @ X6 )
       => ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs2 ) @ X6 )
          = ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7232_sum__count__set,axiom,
    ! [Xs2: list_Extended_enat,X6: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ X6 )
     => ( ( finite4001608067531595151d_enat @ X6 )
       => ( ( groups2027974829824023292at_nat @ ( count_101369445342291426d_enat @ Xs2 ) @ X6 )
          = ( size_s3941691890525107288d_enat @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7233_sum__count__set,axiom,
    ! [Xs2: list_VEBT_VEBT,X6: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ X6 )
     => ( ( finite5795047828879050333T_VEBT @ X6 )
       => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs2 ) @ X6 )
          = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7234_sum__count__set,axiom,
    ! [Xs2: list_o,X6: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ X6 )
     => ( ( finite_finite_o @ X6 )
       => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs2 ) @ X6 )
          = ( size_size_list_o @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7235_sum__count__set,axiom,
    ! [Xs2: list_nat,X6: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ X6 )
     => ( ( finite_finite_nat @ X6 )
       => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs2 ) @ X6 )
          = ( size_size_list_nat @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_7236_one__diff__power__eq,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7237_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7238_one__diff__power__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7239_one__diff__power__eq,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7240_power__diff__1__eq,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7241_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7242_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7243_power__diff__1__eq,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7244_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_7245_geometric__sum,axiom,
    ! [X: complex,N2: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_7246_geometric__sum,axiom,
    ! [X: real,N2: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_7247_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7248_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7249_real__le__lsqrt,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 @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_7250_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_7251_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_7252_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_7253_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_7254_sum__less__suminf,axiom,
    ! [F: nat > int,N2: nat] :
      ( ( summable_int @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7255_sum__less__suminf,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7256_sum__less__suminf,axiom,
    ! [F: nat > real,N2: nat] :
      ( ( summable_real @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7257_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7258_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7259_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7260_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7261_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7262_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7263_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7264_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri2265585572941072030t_real @ N2 )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7265_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7266_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri5044797733671781792omplex @ N2 )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7267_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_7268_sum__gp__strict,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ N2 ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7269_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7270_sum__gp__strict,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ N2 ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7271_lemma__termdiff1,axiom,
    ! [Z2: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z2 @ P5 ) ) @ ( power_power_complex @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7272_lemma__termdiff1,axiom,
    ! [Z2: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z2 @ P5 ) ) @ ( power_power_rat @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7273_lemma__termdiff1,axiom,
    ! [Z2: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z2 @ P5 ) ) @ ( power_power_int @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7274_lemma__termdiff1,axiom,
    ! [Z2: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z2 @ P5 ) ) @ ( power_power_real @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7275_power__diff__sumr2,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7276_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7277_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7278_power__diff__sumr2,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7279_diff__power__eq__sum,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7280_diff__power__eq__sum,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7281_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7282_diff__power__eq__sum,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7283_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_7284_sqrt__sum__squares__le__sum,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 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_7285_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_7286_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( 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 @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7287_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > int,K5: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( 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 @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7288_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( 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 @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7289_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > real,K5: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( 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 @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7290_sum__less__suminf2,axiom,
    ! [F: nat > int,N2: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7291_sum__less__suminf2,axiom,
    ! [F: nat > nat,N2: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7292_sum__less__suminf2,axiom,
    ! [F: nat > real,N2: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7293_one__diff__power__eq_H,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7294_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7295_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7296_one__diff__power__eq_H,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7297_termdiff__converges,axiom,
    ! [X: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
     => ( ! [X4: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
           => ( summable_real
              @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X4 @ N3 ) ) ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_7298_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
     => ( ! [X4: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
           => ( summable_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X4 @ N3 ) ) ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_7299_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_7300_real__sqrt__power__even,axiom,
    ! [N2: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N2 )
          = ( power_power_real @ X @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_7301_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_7302_arith__geo__mean__sqrt,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 @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_7303_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_7304_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_7305_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( 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 @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_7306_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_7307_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_7308_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X3: int] : X3
          @ ( set_or1266510415728281911st_int @ M @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_7309_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_7310_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( 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 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( 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.simps(3)
thf(fact_7311_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_7312_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T3: real] :
            ( ( ord_less_real @ X @ T3 )
            & ( ord_less_real @ T3 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_7313_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_7314_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_7315_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T3: real] :
          ( ( ord_less_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_7316_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_7317_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_7318_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_7319_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_7320_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_7321_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_7322_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_7323_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_7324_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_7325_sin__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( sin_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_7326_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_7327_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_7328_cos__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( cos_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_7329_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_7330_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_7331_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7332_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7333_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7334_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7335_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_7336_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% choose_dvd
thf(fact_7337_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% choose_dvd
thf(fact_7338_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% choose_dvd
thf(fact_7339_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% choose_dvd
thf(fact_7340_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_7341_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_7342_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_7343_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_7344_Maclaurin__exp__lt,axiom,
    ! [X: real,N2: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
            & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_7345_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X7: nat > real] :
          ( ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_real @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) )
          | ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7346_monoseq__def,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X7: nat > set_nat] :
          ( ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_set_nat @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) )
          | ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_set_nat @ ( X7 @ N3 ) @ ( X7 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7347_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X7: nat > rat] :
          ( ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_rat @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) )
          | ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7348_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X7: nat > num] :
          ( ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_num @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) )
          | ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7349_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X7: nat > nat] :
          ( ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_nat @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) )
          | ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7350_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X7: nat > int] :
          ( ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_int @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) )
          | ! [M6: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M6 @ N3 )
             => ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7351_monoI2,axiom,
    ! [X6: nat > real] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_real @ ( X6 @ N ) @ ( X6 @ M5 ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% monoI2
thf(fact_7352_monoI2,axiom,
    ! [X6: nat > set_nat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_set_nat @ ( X6 @ N ) @ ( X6 @ M5 ) ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% monoI2
thf(fact_7353_monoI2,axiom,
    ! [X6: nat > rat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_rat @ ( X6 @ N ) @ ( X6 @ M5 ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% monoI2
thf(fact_7354_monoI2,axiom,
    ! [X6: nat > num] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_num @ ( X6 @ N ) @ ( X6 @ M5 ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% monoI2
thf(fact_7355_monoI2,axiom,
    ! [X6: nat > nat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_nat @ ( X6 @ N ) @ ( X6 @ M5 ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% monoI2
thf(fact_7356_monoI2,axiom,
    ! [X6: nat > int] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_int @ ( X6 @ N ) @ ( X6 @ M5 ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% monoI2
thf(fact_7357_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_7358_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_7359_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_7360_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_7361_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_7362_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_7363_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_7364_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_7365_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_7366_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_7367_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_7368_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_7369_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_7370_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_7371_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_7372_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_7373_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_7374_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_7375_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_7376_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_7377_exp__not__eq__zero,axiom,
    ! [X: real] :
      ( ( exp_real @ X )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_7378_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_7379_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_7380_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X4: real] :
          ( ( exp_real @ X4 )
          = Y ) ) ).

% exp_total
thf(fact_7381_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_7382_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_7383_mult__exp__exp,axiom,
    ! [X: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7384_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
        = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7385_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_7386_exp__minus__inverse,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_7387_exp__minus__inverse,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_7388_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_7389_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X4 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_7390_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_7391_exp__divide__power__eq,axiom,
    ! [N2: nat,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
        = ( exp_complex @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7392_exp__divide__power__eq,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
        = ( exp_real @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7393_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7394_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7395_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_int_def
thf(fact_7396_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_7397_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_7398_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_7399_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_7400_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_7401_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_7402_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_7403_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_7404_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7405_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7406_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_7407_mono__SucI1,axiom,
    ! [X6: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% mono_SucI1
thf(fact_7408_mono__SucI1,axiom,
    ! [X6: nat > set_nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% mono_SucI1
thf(fact_7409_mono__SucI1,axiom,
    ! [X6: nat > rat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% mono_SucI1
thf(fact_7410_mono__SucI1,axiom,
    ! [X6: nat > num] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% mono_SucI1
thf(fact_7411_mono__SucI1,axiom,
    ! [X6: nat > nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% mono_SucI1
thf(fact_7412_mono__SucI1,axiom,
    ! [X6: nat > int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% mono_SucI1
thf(fact_7413_mono__SucI2,axiom,
    ! [X6: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% mono_SucI2
thf(fact_7414_mono__SucI2,axiom,
    ! [X6: nat > set_nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% mono_SucI2
thf(fact_7415_mono__SucI2,axiom,
    ! [X6: nat > rat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% mono_SucI2
thf(fact_7416_mono__SucI2,axiom,
    ! [X6: nat > num] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% mono_SucI2
thf(fact_7417_mono__SucI2,axiom,
    ! [X6: nat > nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% mono_SucI2
thf(fact_7418_mono__SucI2,axiom,
    ! [X6: nat > int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% mono_SucI2
thf(fact_7419_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X7: nat > real] :
          ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7420_monoseq__Suc,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X7: nat > set_nat] :
          ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_set_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7421_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X7: nat > rat] :
          ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7422_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X7: nat > num] :
          ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7423_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X7: nat > nat] :
          ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7424_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X7: nat > int] :
          ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7425_monoI1,axiom,
    ! [X6: nat > real] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_real @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% monoI1
thf(fact_7426_monoI1,axiom,
    ! [X6: nat > set_nat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_set_nat @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% monoI1
thf(fact_7427_monoI1,axiom,
    ! [X6: nat > rat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_rat @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% monoI1
thf(fact_7428_monoI1,axiom,
    ! [X6: nat > num] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_num @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% monoI1
thf(fact_7429_monoI1,axiom,
    ! [X6: nat > nat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_nat @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% monoI1
thf(fact_7430_monoI1,axiom,
    ! [X6: nat > int] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
         => ( ord_less_eq_int @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% monoI1
thf(fact_7431_minus__one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_7432_one__div__minus__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_7433_minus__numeral__div__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_7434_numeral__div__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_7435_pochhammer__double,axiom,
    ! [Z2: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_7436_pochhammer__double,axiom,
    ! [Z2: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_7437_pochhammer__double,axiom,
    ! [Z2: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_7438_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_7439_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_7440_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_7441_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_7442_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_7443_pochhammer__pos,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7444_pochhammer__pos,axiom,
    ! [X: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7445_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7446_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7447_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N2 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7448_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7449_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7450_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7451_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_7452_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_7453_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_7454_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_7455_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_7456_pochhammer__nonneg,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7457_pochhammer__nonneg,axiom,
    ! [X: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7458_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7459_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7460_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_7461_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_7462_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_7463_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_7464_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_7465_pochhammer__rec,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_7466_pochhammer__rec,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_7467_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_7468_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_7469_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_7470_pochhammer__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_7471_pochhammer__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_7472_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_7473_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_7474_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_7475_pochhammer__rec_H,axiom,
    ! [Z2: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_7476_pochhammer__rec_H,axiom,
    ! [Z2: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_7477_pochhammer__rec_H,axiom,
    ! [Z2: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_7478_pochhammer__rec_H,axiom,
    ! [Z2: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_7479_pochhammer__rec_H,axiom,
    ! [Z2: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_7480_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7481_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7482_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7483_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7484_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7485_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7486_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7487_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7488_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7489_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7490_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7491_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7492_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7493_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7494_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7495_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7496_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7497_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7498_pochhammer__product_H,axiom,
    ! [Z2: complex,N2: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7499_pochhammer__product_H,axiom,
    ! [Z2: real,N2: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7500_pochhammer__product_H,axiom,
    ! [Z2: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7501_pochhammer__product_H,axiom,
    ! [Z2: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7502_pochhammer__product_H,axiom,
    ! [Z2: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7503_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s2602460028002588243omplex @ Z2 @ N2 )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7504_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z2 @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7505_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z2 @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7506_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z2 @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7507_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z2 @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7508_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7509_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7510_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7511_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7512_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7513_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% pochhammer_same
thf(fact_7514_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% pochhammer_same
thf(fact_7515_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% pochhammer_same
thf(fact_7516_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% pochhammer_same
thf(fact_7517_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% pochhammer_same
thf(fact_7518_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7519_pochhammer__minus,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7520_pochhammer__minus,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7521_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7522_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7523_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7524_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7525_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7526_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7527_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7528_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_double
thf(fact_7529_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_double
thf(fact_7530_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_double
thf(fact_7531_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: complex,N2: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7532_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7533_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7534_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A2: complex,N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A2 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_7535_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A2: real,N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A2 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_7536_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A2: rat,N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_7537_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A2: int,N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A2 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_7538_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A2: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A2 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_7539_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N3: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
          @ N3
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_7540_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N3: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N3
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_7541_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N3: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N3
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_7542_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N3
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_7543_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N3
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_7544_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7545_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7546_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7547_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_7548_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_7549_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_7550_Ints__prod,axiom,
    ! [A3: set_complex,F: complex > int] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( member_int @ ( F @ X4 ) @ ring_1_Ints_int ) )
     => ( member_int @ ( groups858564598930262913ex_int @ F @ A3 ) @ ring_1_Ints_int ) ) ).

% Ints_prod
thf(fact_7551_Ints__prod,axiom,
    ! [A3: set_real,F: real > int] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( member_int @ ( F @ X4 ) @ ring_1_Ints_int ) )
     => ( member_int @ ( groups4694064378042380927al_int @ F @ A3 ) @ ring_1_Ints_int ) ) ).

% Ints_prod
thf(fact_7552_Ints__prod,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_prod
thf(fact_7553_Ints__prod,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_prod
thf(fact_7554_Ints__prod,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups129246275422532515t_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_prod
thf(fact_7555_Ints__prod,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( member_real @ ( F @ X4 ) @ ring_1_Ints_real ) )
     => ( member_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ring_1_Ints_real ) ) ).

% Ints_prod
thf(fact_7556_Ints__prod,axiom,
    ! [A3: set_complex,F: complex > complex] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups3708469109370488835omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_prod
thf(fact_7557_Ints__prod,axiom,
    ! [A3: set_real,F: real > complex] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups713298508707869441omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_prod
thf(fact_7558_Ints__prod,axiom,
    ! [A3: set_nat,F: nat > complex] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups6464643781859351333omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_prod
thf(fact_7559_Ints__prod,axiom,
    ! [A3: set_int,F: int > complex] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( member_complex @ ( F @ X4 ) @ ring_1_Ints_complex ) )
     => ( member_complex @ ( groups7440179247065528705omplex @ F @ A3 ) @ ring_1_Ints_complex ) ) ).

% Ints_prod
thf(fact_7560_prod_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A3 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_7561_prod_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A3 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7562_prod_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A3 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7563_of__nat__prod,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_7564_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups6464643781859351333omplex
        @ ^ [X3: nat] : ( semiri8010041392384452111omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_7565_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups129246275422532515t_real
        @ ^ [X3: nat] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_7566_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X3: nat] : ( semiri681578069525770553at_rat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_7567_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_7568_of__nat__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_nat_prod
thf(fact_7569_prod__zero__iff,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups129246275422532515t_real @ F @ A3 )
          = zero_zero_real )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7570_prod__zero__iff,axiom,
    ! [A3: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups2316167850115554303t_real @ F @ A3 )
          = zero_zero_real )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7571_prod__zero__iff,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups766887009212190081x_real @ F @ A3 )
          = zero_zero_real )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7572_prod__zero__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups97031904164794029t_real @ F @ A3 )
          = zero_zero_real )
        = ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7573_prod__zero__iff,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups73079841787564623at_rat @ F @ A3 )
          = zero_zero_rat )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7574_prod__zero__iff,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups1072433553688619179nt_rat @ F @ A3 )
          = zero_zero_rat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7575_prod__zero__iff,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups225925009352817453ex_rat @ F @ A3 )
          = zero_zero_rat )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7576_prod__zero__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups2245840878043517529at_rat @ F @ A3 )
          = zero_zero_rat )
        = ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7577_prod__zero__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A3 )
          = zero_zero_nat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7578_prod__zero__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A3 )
          = zero_zero_nat )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
              & ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_7579_prod_Oempty,axiom,
    ! [G: extended_enat > complex] :
      ( ( groups4622424608036095791omplex @ G @ bot_bo7653980558646680370d_enat )
      = one_one_complex ) ).

% prod.empty
thf(fact_7580_prod_Oempty,axiom,
    ! [G: extended_enat > real] :
      ( ( groups97031904164794029t_real @ G @ bot_bo7653980558646680370d_enat )
      = one_one_real ) ).

% prod.empty
thf(fact_7581_prod_Oempty,axiom,
    ! [G: extended_enat > rat] :
      ( ( groups2245840878043517529at_rat @ G @ bot_bo7653980558646680370d_enat )
      = one_one_rat ) ).

% prod.empty
thf(fact_7582_prod_Oempty,axiom,
    ! [G: extended_enat > nat] :
      ( ( groups2880970938130013265at_nat @ G @ bot_bo7653980558646680370d_enat )
      = one_one_nat ) ).

% prod.empty
thf(fact_7583_prod_Oempty,axiom,
    ! [G: extended_enat > int] :
      ( ( groups2878480467620962989at_int @ G @ bot_bo7653980558646680370d_enat )
      = one_one_int ) ).

% prod.empty
thf(fact_7584_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_7585_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_7586_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_7587_prod_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_7588_prod_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_7589_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups6464643781859351333omplex @ G @ A3 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7590_prod_Oinfinite,axiom,
    ! [A3: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups7440179247065528705omplex @ G @ A3 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7591_prod_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups3708469109370488835omplex @ G @ A3 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7592_prod_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > complex] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4622424608036095791omplex @ G @ A3 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7593_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups129246275422532515t_real @ G @ A3 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7594_prod_Oinfinite,axiom,
    ! [A3: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups2316167850115554303t_real @ G @ A3 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7595_prod_Oinfinite,axiom,
    ! [A3: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups766887009212190081x_real @ G @ A3 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7596_prod_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups97031904164794029t_real @ G @ A3 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7597_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups73079841787564623at_rat @ G @ A3 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7598_prod_Oinfinite,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups1072433553688619179nt_rat @ G @ A3 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7599_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_7600_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_7601_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_7602_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_7603_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_7604_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_7605_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_7606_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_7607_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_7608_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_7609_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_7610_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_7611_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_7612_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_7613_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7614_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7615_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7616_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7617_prod_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7618_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7619_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7620_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7621_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7622_prod_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7623_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7624_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7625_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7626_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7627_prod_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4622424608036095791omplex
              @ ^ [K3: extended_enat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7628_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7629_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7630_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7631_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7632_prod_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups97031904164794029t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7633_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq_nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_7634_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_7635_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_7636_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_7637_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_7638_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7639_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7640_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7641_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7642_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7643_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > complex,A3: set_complex] :
      ( ( ( groups3708469109370488835omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A4: complex] :
            ( ( member_complex @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7644_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A3: set_real] :
      ( ( ( groups713298508707869441omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7645_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A3: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7646_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A3: set_int] :
      ( ( ( groups7440179247065528705omplex @ G @ A3 )
       != one_one_complex )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7647_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A3: set_complex] :
      ( ( ( groups766887009212190081x_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A4: complex] :
            ( ( member_complex @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7648_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A3: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7649_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A3: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7650_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A3: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A3 )
       != one_one_real )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7651_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > rat,A3: set_complex] :
      ( ( ( groups225925009352817453ex_rat @ G @ A3 )
       != one_one_rat )
     => ~ ! [A4: complex] :
            ( ( member_complex @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7652_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A3: set_real] :
      ( ( ( groups4061424788464935467al_rat @ G @ A3 )
       != one_one_rat )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A3 )
           => ( ( G @ A4 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7653_prod_Oneutral,axiom,
    ! [A3: set_nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ( G @ X4 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A3 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_7654_prod_Oneutral,axiom,
    ! [A3: set_nat,G: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ( G @ X4 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A3 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_7655_prod_Oneutral,axiom,
    ! [A3: set_int,G: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ( G @ X4 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A3 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_7656_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_7657_prod__mono,axiom,
    ! [A3: set_complex,F: complex > real,G: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ( groups766887009212190081x_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7658_prod__mono,axiom,
    ! [A3: set_real,F: real > real,G: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ( groups1681761925125756287l_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7659_prod__mono,axiom,
    ! [A3: set_nat,F: nat > real,G: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A3 ) @ ( groups129246275422532515t_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7660_prod__mono,axiom,
    ! [A3: set_int,F: int > real,G: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ( groups2316167850115554303t_real @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7661_prod__mono,axiom,
    ! [A3: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ ( groups225925009352817453ex_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7662_prod__mono,axiom,
    ! [A3: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ ( groups4061424788464935467al_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7663_prod__mono,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7664_prod__mono,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7665_prod__mono,axiom,
    ! [A3: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A3 ) @ ( groups861055069439313189ex_nat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7666_prod__mono,axiom,
    ! [A3: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A3 ) @ ( groups4696554848551431203al_nat @ G @ A3 ) ) ) ).

% prod_mono
thf(fact_7667_prod__nonneg,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_7668_prod__nonneg,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_7669_prod__nonneg,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).

% prod_nonneg
thf(fact_7670_prod__pos,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_7671_prod__pos,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_7672_prod__pos,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A3 ) ) ) ).

% prod_pos
thf(fact_7673_prod__ge__1,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7674_prod__ge__1,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7675_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7676_prod__ge__1,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7677_prod__ge__1,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7678_prod__ge__1,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7679_prod__ge__1,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7680_prod__ge__1,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7681_prod__ge__1,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups861055069439313189ex_nat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7682_prod__ge__1,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A3 ) ) ) ).

% prod_ge_1
thf(fact_7683_prod__zero,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups129246275422532515t_real @ F @ A3 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7684_prod__zero,axiom,
    ! [A3: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups2316167850115554303t_real @ F @ A3 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7685_prod__zero,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups766887009212190081x_real @ F @ A3 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7686_prod__zero,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ? [X2: extended_enat] :
            ( ( member_Extended_enat @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_real ) )
       => ( ( groups97031904164794029t_real @ F @ A3 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_7687_prod__zero,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups73079841787564623at_rat @ F @ A3 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7688_prod__zero,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups1072433553688619179nt_rat @ F @ A3 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7689_prod__zero,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups225925009352817453ex_rat @ F @ A3 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7690_prod__zero,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ? [X2: extended_enat] :
            ( ( member_Extended_enat @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_rat ) )
       => ( ( groups2245840878043517529at_rat @ F @ A3 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_7691_prod__zero,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_nat ) )
       => ( ( groups1707563613775114915nt_nat @ F @ A3 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_7692_prod__zero,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A3 )
            & ( ( F @ X2 )
              = zero_zero_nat ) )
       => ( ( groups861055069439313189ex_nat @ F @ A3 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_7693_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_7694_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A2: nat] : ( times_times_complex @ ( F @ A2 ) )
        @ A
        @ B
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_7695_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A2: nat] : ( times_times_real @ ( F @ A2 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_7696_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A2: nat] : ( times_times_rat @ ( F @ A2 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7697_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A2: nat] : ( times_times_nat @ ( F @ A2 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7698_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A2: nat] : ( times_times_int @ ( F @ A2 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_7699_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_7700_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_7701_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_7702_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_7703_binomial__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_7704_prod_Ointer__filter,axiom,
    ! [A3: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups713298508707869441omplex @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups713298508707869441omplex
          @ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7705_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups6464643781859351333omplex @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups6464643781859351333omplex
          @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7706_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7440179247065528705omplex @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups7440179247065528705omplex
          @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7707_prod_Ointer__filter,axiom,
    ! [A3: set_complex,G: complex > complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups3708469109370488835omplex
          @ ^ [X3: complex] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7708_prod_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > complex,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4622424608036095791omplex @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups4622424608036095791omplex
          @ ^ [X3: extended_enat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_complex )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7709_prod_Ointer__filter,axiom,
    ! [A3: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups1681761925125756287l_real
          @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7710_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups129246275422532515t_real @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X3: nat] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7711_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7712_prod_Ointer__filter,axiom,
    ! [A3: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups766887009212190081x_real @ G
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups766887009212190081x_real
          @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7713_prod_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > real,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups97031904164794029t_real @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A3 )
                & ( P @ X3 ) ) ) )
        = ( groups97031904164794029t_real
          @ ^ [X3: extended_enat] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_real )
          @ A3 ) ) ) ).

% prod.inter_filter
thf(fact_7714_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7715_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7716_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7717_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7718_prod__le__1,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
            & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7719_prod__le__1,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7720_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7721_prod__le__1,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7722_prod__le__1,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7723_prod__le__1,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7724_prod__le__1,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7725_prod__le__1,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7726_prod__le__1,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A3 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7727_prod__le__1,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A3 )
         => ( ( 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 @ A3 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7728_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups6464643781859351333omplex @ H2 @ S3 ) @ ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7729_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups7440179247065528705omplex @ H2 @ S3 ) @ ( groups7440179247065528705omplex @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7730_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups3708469109370488835omplex @ H2 @ S3 ) @ ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7731_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_Extended_enat,H2: extended_enat > complex,G: extended_enat > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups4622424608036095791omplex @ H2 @ S3 ) @ ( groups4622424608036095791omplex @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7732_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_nat,H2: nat > real,G: nat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups129246275422532515t_real @ H2 @ S3 ) @ ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7733_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups2316167850115554303t_real @ H2 @ S3 ) @ ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7734_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups766887009212190081x_real @ H2 @ S3 ) @ ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7735_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups97031904164794029t_real @ H2 @ S3 ) @ ( groups97031904164794029t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7736_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups73079841787564623at_rat @ H2 @ S3 ) @ ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7737_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups1072433553688619179nt_rat @ H2 @ S3 ) @ ( groups1072433553688619179nt_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7738_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S3: set_real,I: real > real,J: real > real,T5: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7739_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S3: set_real,I: int > real,J: real > int,T5: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7740_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S3: set_real,I: complex > real,J: real > complex,T5: set_complex,G: real > complex,H2: complex > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7741_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_Extended_enat,S3: set_real,I: extended_enat > real,J: real > extended_enat,T5: set_Extended_enat,G: real > complex,H2: extended_enat > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite4001608067531595151d_enat @ T4 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T5 @ T4 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                   => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups4622424608036095791omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7742_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S3: set_int,I: real > int,J: int > real,T5: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7743_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S3: set_int,I: int > int,J: int > int,T5: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7744_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,S3: set_int,I: complex > int,J: int > complex,T5: set_complex,G: int > complex,H2: complex > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7745_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_Extended_enat,S3: set_int,I: extended_enat > int,J: int > extended_enat,T5: set_Extended_enat,G: int > complex,H2: extended_enat > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite4001608067531595151d_enat @ T4 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_Extended_enat @ ( J @ A4 ) @ ( minus_925952699566721837d_enat @ T5 @ T4 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups4622424608036095791omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7746_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_real,S3: set_complex,I: real > complex,J: complex > real,T5: set_real,G: complex > complex,H2: real > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S3 )
                        = ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7747_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_int,S3: set_complex,I: int > complex,J: complex > int,T5: set_int,G: complex > complex,H2: int > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A4: complex] :
              ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_complex @ ( I @ B4 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A4: complex] :
                      ( ( member_complex @ A4 @ S4 )
                     => ( ( G @ A4 )
                        = one_one_complex ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = one_one_complex ) )
                   => ( ! [A4: complex] :
                          ( ( member_complex @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S3 )
                        = ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7748_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_7749_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_7750_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_7751_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_7752_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_7753_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7754_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7755_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7756_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N2 ) @ K )
        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7757_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7758_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K )
        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7759_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: real > complex] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups713298508707869441omplex @ G
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X3: real] :
                  ( ( G @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups713298508707869441omplex @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7760_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > complex] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups7440179247065528705omplex @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups7440179247065528705omplex @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7761_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( minus_811609699411566653omplex @ A3
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups3708469109370488835omplex @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7762_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4622424608036095791omplex @ G
          @ ( minus_925952699566721837d_enat @ A3
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G @ X3 )
                  = one_one_complex ) ) ) )
        = ( groups4622424608036095791omplex @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7763_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X3: real] :
                  ( ( G @ X3 )
                  = one_one_real ) ) ) )
        = ( groups1681761925125756287l_real @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7764_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_real ) ) ) )
        = ( groups2316167850115554303t_real @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7765_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups766887009212190081x_real @ G
          @ ( minus_811609699411566653omplex @ A3
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = one_one_real ) ) ) )
        = ( groups766887009212190081x_real @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7766_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups97031904164794029t_real @ G
          @ ( minus_925952699566721837d_enat @ A3
            @ ( collec4429806609662206161d_enat
              @ ^ [X3: extended_enat] :
                  ( ( G @ X3 )
                  = one_one_real ) ) ) )
        = ( groups97031904164794029t_real @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7767_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X3: real] :
                  ( ( G @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups4061424788464935467al_rat @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7768_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7769_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_7770_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_7771_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7772_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7773_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7774_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7775_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7776_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7777_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7778_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7779_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7780_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7781_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7782_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7783_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7784_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7785_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7786_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7787_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7788_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7789_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7790_less__1__prod,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7791_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7792_less__1__prod,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7793_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7794_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2245840878043517529at_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7795_less__1__prod,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7796_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7797_less__1__prod,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7798_prod_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ A3 )
                  = ( groups713298508707869441omplex @ H2 @ B3 ) )
                = ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7799_prod_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ A3 )
                  = ( groups7440179247065528705omplex @ H2 @ B3 ) )
                = ( ( groups7440179247065528705omplex @ G @ C4 )
                  = ( groups7440179247065528705omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7800_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ A3 )
                  = ( groups3708469109370488835omplex @ H2 @ B3 ) )
                = ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7801_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > complex,H2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups4622424608036095791omplex @ G @ A3 )
                  = ( groups4622424608036095791omplex @ H2 @ B3 ) )
                = ( ( groups4622424608036095791omplex @ G @ C4 )
                  = ( groups4622424608036095791omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7802_prod_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A3 )
                  = ( groups1681761925125756287l_real @ H2 @ B3 ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7803_prod_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ A3 )
                  = ( groups2316167850115554303t_real @ H2 @ B3 ) )
                = ( ( groups2316167850115554303t_real @ G @ C4 )
                  = ( groups2316167850115554303t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7804_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ A3 )
                  = ( groups766887009212190081x_real @ H2 @ B3 ) )
                = ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7805_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups97031904164794029t_real @ G @ A3 )
                  = ( groups97031904164794029t_real @ H2 @ B3 ) )
                = ( ( groups97031904164794029t_real @ G @ C4 )
                  = ( groups97031904164794029t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7806_prod_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ A3 )
                  = ( groups4061424788464935467al_rat @ H2 @ B3 ) )
                = ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7807_prod_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ A3 )
                  = ( groups1072433553688619179nt_rat @ H2 @ B3 ) )
                = ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7808_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) )
               => ( ( groups713298508707869441omplex @ G @ A3 )
                  = ( groups713298508707869441omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7809_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ C4 )
                  = ( groups7440179247065528705omplex @ H2 @ C4 ) )
               => ( ( groups7440179247065528705omplex @ G @ A3 )
                  = ( groups7440179247065528705omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7810_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) )
               => ( ( groups3708469109370488835omplex @ G @ A3 )
                  = ( groups3708469109370488835omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7811_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > complex,H2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_complex ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_complex ) )
             => ( ( ( groups4622424608036095791omplex @ G @ C4 )
                  = ( groups4622424608036095791omplex @ H2 @ C4 ) )
               => ( ( groups4622424608036095791omplex @ G @ A3 )
                  = ( groups4622424608036095791omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7812_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A3 )
                  = ( groups1681761925125756287l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7813_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ C4 )
                  = ( groups2316167850115554303t_real @ H2 @ C4 ) )
               => ( ( groups2316167850115554303t_real @ G @ A3 )
                  = ( groups2316167850115554303t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7814_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) )
               => ( ( groups766887009212190081x_real @ G @ A3 )
                  = ( groups766887009212190081x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7815_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A3: set_Extended_enat,B3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B3 @ C4 )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ ( minus_925952699566721837d_enat @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups97031904164794029t_real @ G @ C4 )
                  = ( groups97031904164794029t_real @ H2 @ C4 ) )
               => ( ( groups97031904164794029t_real @ G @ A3 )
                  = ( groups97031904164794029t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7816_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A3 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
               => ( ( groups4061424788464935467al_rat @ G @ A3 )
                  = ( groups4061424788464935467al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7817_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A3 @ C4 )
       => ( ( ord_less_eq_set_int @ B3 @ C4 )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ C4 @ A3 ) )
               => ( ( G @ A4 )
                  = one_one_rat ) )
           => ( ! [B4: int] :
                  ( ( member_int @ B4 @ ( minus_minus_set_int @ C4 @ B3 ) )
                 => ( ( H2 @ B4 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ C4 ) )
               => ( ( groups1072433553688619179nt_rat @ G @ A3 )
                  = ( groups1072433553688619179nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7818_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G @ S3 )
            = ( groups7440179247065528705omplex @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7819_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ S3 )
            = ( groups3708469109370488835omplex @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7820_prod_Omono__neutral__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ( groups4622424608036095791omplex @ G @ S3 )
            = ( groups4622424608036095791omplex @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7821_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G @ S3 )
            = ( groups2316167850115554303t_real @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7822_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S3 )
            = ( groups766887009212190081x_real @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7823_prod_Omono__neutral__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G @ S3 )
            = ( groups97031904164794029t_real @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7824_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G @ S3 )
            = ( groups1072433553688619179nt_rat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7825_prod_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ S3 )
            = ( groups225925009352817453ex_rat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7826_prod_Omono__neutral__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups2245840878043517529at_rat @ G @ S3 )
            = ( groups2245840878043517529at_rat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7827_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ S3 )
            = ( groups1707563613775114915nt_nat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7828_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G @ T5 )
            = ( groups7440179247065528705omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7829_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ T5 )
            = ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7830_prod_Omono__neutral__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ( groups4622424608036095791omplex @ G @ T5 )
            = ( groups4622424608036095791omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7831_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G @ T5 )
            = ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7832_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T5 )
            = ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7833_prod_Omono__neutral__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G @ T5 )
            = ( groups97031904164794029t_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7834_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G @ T5 )
            = ( groups1072433553688619179nt_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7835_prod_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ T5 )
            = ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7836_prod_Omono__neutral__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups2245840878043517529at_rat @ G @ T5 )
            = ( groups2245840878043517529at_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7837_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ T5 )
            = ( groups1707563613775114915nt_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7838_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups713298508707869441omplex @ G @ S3 )
              = ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7839_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ S3 )
              = ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7840_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ S3 )
              = ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7841_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > complex,G: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4622424608036095791omplex @ G @ S3 )
              = ( groups4622424608036095791omplex @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7842_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S3 )
              = ( groups1681761925125756287l_real @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7843_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ S3 )
              = ( groups2316167850115554303t_real @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7844_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups766887009212190081x_real @ G @ S3 )
              = ( groups766887009212190081x_real @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7845_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_real ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups97031904164794029t_real @ G @ S3 )
              = ( groups97031904164794029t_real @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7846_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ S3 )
              = ( groups4061424788464935467al_rat @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7847_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H2 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ S3 )
              = ( groups1072433553688619179nt_rat @ H2 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7848_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups713298508707869441omplex @ G @ T5 )
              = ( groups713298508707869441omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7849_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ T5 )
              = ( groups7440179247065528705omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7850_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ T5 )
              = ( groups3708469109370488835omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7851_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > complex,H2: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4622424608036095791omplex @ G @ T5 )
              = ( groups4622424608036095791omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7852_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T5 )
              = ( groups1681761925125756287l_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7853_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ T5 )
              = ( groups2316167850115554303t_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7854_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups766887009212190081x_real @ G @ T5 )
              = ( groups766887009212190081x_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7855_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T5 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T5 )
       => ( ! [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_real ) )
         => ( ! [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups97031904164794029t_real @ G @ T5 )
              = ( groups97031904164794029t_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7856_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ T5 )
              = ( groups4061424788464935467al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7857_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ T5 )
              = ( groups1072433553688619179nt_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7858_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7859_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7860_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7861_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7862_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7863_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7864_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7865_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7866_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7867_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7868_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7869_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7870_binomial__absorption,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_7871_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7872_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7873_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7874_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7875_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7876_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7877_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7878_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7879_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7880_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7881_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7882_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7883_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7884_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7885_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_7886_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7887_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7888_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7889_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7890_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ M )
          @ ( groups129246275422532515t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7891_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_rat @ ( G @ M )
          @ ( groups73079841787564623at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7892_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7893_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7894_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_7895_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_7896_fact__prod,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N3: nat] :
          ( semiri681578069525770553at_rat
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7897_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N3: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7898_fact__prod,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N3: nat] :
          ( semiri5074537144036343181t_real
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7899_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N3: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7900_fact__prod,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N3: nat] :
          ( semiri8010041392384452111omplex
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7901_prod__mono__strict,axiom,
    ! [A3: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A3 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ( groups766887009212190081x_real @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7902_prod__mono__strict,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ A3 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => ( ord_less_real @ ( groups97031904164794029t_real @ F @ A3 ) @ ( groups97031904164794029t_real @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7903_prod__mono__strict,axiom,
    ! [A3: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A3 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ( groups1681761925125756287l_real @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7904_prod__mono__strict,axiom,
    ! [A3: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A3 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A3 ) @ ( groups129246275422532515t_real @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7905_prod__mono__strict,axiom,
    ! [A3: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A3 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ( groups2316167850115554303t_real @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7906_prod__mono__strict,axiom,
    ! [A3: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ ( groups225925009352817453ex_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7907_prod__mono__strict,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,G: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => ( ord_less_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ ( groups2245840878043517529at_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7908_prod__mono__strict,axiom,
    ! [A3: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_real )
         => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ ( groups4061424788464935467al_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7909_prod__mono__strict,axiom,
    ! [A3: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A3 ) @ ( groups73079841787564623at_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7910_prod__mono__strict,axiom,
    ! [A3: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A3 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A3 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ G @ A3 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7911_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7912_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7913_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% binomial_maximum'
thf(fact_7914_binomial__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_7915_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_7916_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_7917_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_7918_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_7919_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_7920_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7921_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7922_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7923_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7924_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_7925_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F4: nat > nat > nat,A2: nat,B2: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A2 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F4 @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F4 @ A2 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7926_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7927_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7928_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7929_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7930_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7931_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7932_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_7933_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7934_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7935_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7936_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_7937_prod__mono2,axiom,
    ! [B3: set_real,A3: set_real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ A3 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A3 ) @ ( groups1681761925125756287l_real @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7938_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > real] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ A3 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A3 ) @ ( groups2316167850115554303t_real @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7939_prod__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ A3 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A3 ) @ ( groups766887009212190081x_real @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7940_prod__mono2,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B3 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B3 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B3 @ A3 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ A3 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A4 ) ) )
           => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A3 ) @ ( groups97031904164794029t_real @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7941_prod__mono2,axiom,
    ! [B3: set_real,A3: set_real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A3 ) @ ( groups4061424788464935467al_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7942_prod__mono2,axiom,
    ! [B3: set_int,A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( ! [B4: int] :
              ( ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A3 ) @ ( groups1072433553688619179nt_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7943_prod__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A3 ) @ ( groups225925009352817453ex_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7944_prod__mono2,axiom,
    ! [B3: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ B3 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B3 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B3 @ A3 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B4 ) ) )
         => ( ! [A4: extended_enat] :
                ( ( member_Extended_enat @ A4 @ A3 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
           => ( ord_less_eq_rat @ ( groups2245840878043517529at_rat @ F @ A3 ) @ ( groups2245840878043517529at_rat @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7945_prod__mono2,axiom,
    ! [B3: set_real,A3: set_real,F: real > int] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A3 @ B3 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A3 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ A3 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A3 ) @ ( groups4694064378042380927al_int @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7946_prod__mono2,axiom,
    ! [B3: set_complex,A3: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B3 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A3 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ A3 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A3 ) @ ( groups858564598930262913ex_int @ F @ B3 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7947_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_7948_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_7949_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_7950_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_7951_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A2: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7952_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A2: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7953_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A2: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7954_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_7955_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_7956_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_7957_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_7958_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_7959_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_7960_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_7961_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7962_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7963_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7964_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7965_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7966_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7967_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A2: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A2 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7968_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A2: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A2 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7969_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A2: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A2 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7970_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ K ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_7971_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ K ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_7972_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_7973_pochhammer__Suc__prod,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7974_pochhammer__Suc__prod,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7975_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7976_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7977_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7978_pochhammer__prod__rev,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A2: complex,N3: nat] :
          ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A2 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7979_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A2: real,N3: nat] :
          ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A2 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7980_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A2: rat,N3: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7981_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A2: nat,N3: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A2 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7982_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A2: int,N3: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7983_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K3 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_7984_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_7985_choose__two,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_7986_gbinomial__minus,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_7987_gbinomial__minus,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_7988_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_7989_prod_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7990_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7991_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7992_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_7993_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7994_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7995_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7996_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A2: nat] : ( plus_plus_rat @ ( F @ A2 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7997_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A2: nat] : ( plus_plus_int @ ( F @ A2 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_7998_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A2: nat] : ( plus_plus_nat @ ( F @ A2 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7999_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A2: nat] : ( plus_plus_real @ ( F @ A2 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_8000_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8001_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8002_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8003_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8004_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8005_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A2: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A2 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8006_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A2: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A2 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8007_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A2: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A2 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8008_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A2: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8009_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A2: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8010_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A2: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8011_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8012_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8013_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8014_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8015_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8016_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8017_fact__code,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8018_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8019_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8020_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8021_fact__code,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N3: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8022_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8023_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8024_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8025_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8026_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8027_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8028_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8029_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8030_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8031_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8032_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8033_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8034_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8035_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8036_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_8037_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I @ K ) ) ).

% atMost_iff
thf(fact_8038_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
      = ( ord_less_eq_set_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_8039_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I @ K ) ) ).

% atMost_iff
thf(fact_8040_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I @ K ) ) ).

% atMost_iff
thf(fact_8041_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_8042_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_8043_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_8044_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_8045_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_8046_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_8047_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_8048_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_8049_prod__eq__1__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A3 )
          = one_one_nat )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8050_prod__eq__1__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A3 )
          = one_one_nat )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8051_prod__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( groups4077766827762148844at_nat @ F @ A3 )
          = one_one_nat )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A3 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8052_prod__eq__1__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups2880970938130013265at_nat @ F @ A3 )
          = one_one_nat )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A3 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8053_prod__eq__1__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups708209901874060359at_nat @ F @ A3 )
          = one_one_nat )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8054_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H2: set_nat,H3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
      = ( ~ ( ord_less_eq_set_nat @ L @ H2 )
        | ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8055_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_8056_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_8057_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_8058_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_8059_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_8060_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8061_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8062_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8063_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8064_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8065_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8066_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8067_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8068_prod__pos__nat__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A3 ) )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8069_prod__pos__nat__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A3 ) )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8070_prod__pos__nat__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups4077766827762148844at_nat @ F @ A3 ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8071_prod__pos__nat__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups2880970938130013265at_nat @ F @ A3 ) )
        = ( ! [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8072_prod__pos__nat__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A3 ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A3 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8073_int__prod,axiom,
    ! [F: int > nat,A3: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% int_prod
thf(fact_8074_int__prod,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% int_prod
thf(fact_8075_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: extended_enat] :
      ( bot_bo7653980558646680370d_enat
     != ( set_or8332593352340944941d_enat @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_8076_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_8077_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_8078_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_8079_infinite__Iic,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).

% infinite_Iic
thf(fact_8080_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_8081_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_8082_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X3: real] : ( ord_less_eq_real @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8083_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8084_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8085_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X3: num] : ( ord_less_eq_num @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8086_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8087_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8088_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_8089_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8090_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_8091_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_8092_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_8093_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_8094_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_8095_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_8096_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_8097_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_8098_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_8099_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8100_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8101_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8102_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8103_sum__telescope,axiom,
    ! [F: nat > rat,I: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8104_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8105_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8106_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N2: nat,D: nat > complex] :
      ( ( ! [X3: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8107_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N2: nat,D: nat > real] :
      ( ( ! [X3: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8108_bounded__imp__summable,axiom,
    ! [A: nat > int,B3: int] :
      ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N ) ) @ B3 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8109_bounded__imp__summable,axiom,
    ! [A: nat > nat,B3: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N ) ) @ B3 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8110_bounded__imp__summable,axiom,
    ! [A: nat > real,B3: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N ) ) @ B3 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8111_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8112_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8113_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8114_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8115_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8116_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_8117_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_8118_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8119_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8120_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_8121_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_8122_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_8123_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N2: nat,K: nat] :
      ( ! [W: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8124_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N2: nat,K: nat] :
      ( ! [W: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8125_polyfun__eq__0,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( ! [X3: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_complex ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_8126_polyfun__eq__0,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( ! [X3: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_real ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_8127_ln__prod,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I5 ) )
          = ( groups8097168146408367636l_real
            @ ^ [X3: real] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8128_ln__prod,axiom,
    ! [I5: set_set_nat,F: set_nat > real] :
      ( ( finite1152437895449049373et_nat @ I5 )
     => ( ! [I2: set_nat] :
            ( ( member_set_nat @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups3619160379726066777t_real @ F @ I5 ) )
          = ( groups5107569545109728110t_real
            @ ^ [X3: set_nat] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8129_ln__prod,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I5 ) )
          = ( groups8778361861064173332t_real
            @ ^ [X3: int] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8130_ln__prod,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I5 ) )
          = ( groups5808333547571424918x_real
            @ ^ [X3: complex] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8131_ln__prod,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ I5 )
     => ( ! [I2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups6036352826371341000t_real @ F @ I5 ) )
          = ( groups4567486121110086003t_real
            @ ^ [X3: product_prod_nat_nat] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8132_ln__prod,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups97031904164794029t_real @ F @ I5 ) )
          = ( groups4148127829035722712t_real
            @ ^ [X3: extended_enat] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8133_ln__prod,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I5 ) )
          = ( groups6591440286371151544t_real
            @ ^ [X3: nat] : ( ln_ln_real @ ( F @ X3 ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8134_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8135_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8136_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8137_sum_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8138_sum__up__index__split,axiom,
    ! [F: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8139_sum__up__index__split,axiom,
    ! [F: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8140_sum__up__index__split,axiom,
    ! [F: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8141_sum__up__index__split,axiom,
    ! [F: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8142_prod_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8143_prod_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8144_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8145_prod_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8146_gbinomial__parallel__sum,axiom,
    ! [A: complex,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8147_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8148_gbinomial__parallel__sum,axiom,
    ! [A: real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8149_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_8150_vandermonde,axiom,
    ! [M: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_8151_sum__gp__basic,axiom,
    ! [X: complex,N2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8152_sum__gp__basic,axiom,
    ! [X: rat,N2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8153_sum__gp__basic,axiom,
    ! [X: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8154_sum__gp__basic,axiom,
    ! [X: real,N2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8155_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_complex ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
            & ( ( C @ I3 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_8156_polyfun__finite__roots,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X3: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_real ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
            & ( ( C @ I3 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_8157_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_8158_polyfun__roots__finite,axiom,
    ! [C: nat > real,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z5: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_8159_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N2: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex )
     => ~ ! [B4: nat > complex] :
            ~ ! [Z4: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I3: nat] : ( times_times_complex @ ( B4 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8160_polyfun__linear__factor__root,axiom,
    ! [C: nat > rat,A: rat,N2: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat )
     => ~ ! [B4: nat > rat] :
            ~ ! [Z4: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I3: nat] : ( times_times_rat @ ( B4 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8161_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N2: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int )
     => ~ ! [B4: nat > int] :
            ~ ! [Z4: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_int @ ( minus_minus_int @ Z4 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I3: nat] : ( times_times_int @ ( B4 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8162_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N2: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real )
     => ~ ! [B4: nat > real] :
            ~ ! [Z4: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_real @ ( minus_minus_real @ Z4 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( B4 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8163_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N2: nat,A: complex] :
    ? [B4: nat > complex] :
    ! [Z4: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( B4 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8164_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N2: nat,A: rat] :
    ? [B4: nat > rat] :
    ! [Z4: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( B4 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8165_polyfun__linear__factor,axiom,
    ! [C: nat > int,N2: nat,A: int] :
    ? [B4: nat > int] :
    ! [Z4: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( B4 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8166_polyfun__linear__factor,axiom,
    ! [C: nat > real,N2: nat,A: real] :
    ? [B4: nat > real] :
    ! [Z4: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( B4 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8167_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8168_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8169_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8170_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8171_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_8172_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8173_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8174_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8175_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8176_polynomial__product,axiom,
    ! [M: nat,A: nat > complex,N2: nat,B: nat > complex,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( 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 @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8177_polynomial__product,axiom,
    ! [M: nat,A: nat > rat,N2: nat,B: nat > rat,X: rat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( 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 @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8178_polynomial__product,axiom,
    ! [M: nat,A: nat > int,N2: nat,B: nat > int,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( 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 @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8179_polynomial__product,axiom,
    ! [M: nat,A: nat > real,N2: nat,B: nat > real,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( 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 @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8180_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8181_prod_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8182_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8183_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8184_polyfun__eq__const,axiom,
    ! [C: nat > complex,N2: nat,K: complex] :
      ( ( ! [X3: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_8185_polyfun__eq__const,axiom,
    ! [C: nat > real,N2: nat,K: real] :
      ( ( ! [X3: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_8186_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8187_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8188_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8189_binomial__ring,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8190_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8191_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8192_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8193_binomial__ring,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8194_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K3 ) ) @ ( comm_s2602460028002588243omplex @ A @ K3 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8195_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8196_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8197_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8198_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( 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 @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8199_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8200_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8201_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8202_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8203_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8204_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8205_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8206_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8207_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8208_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8209_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8210_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8211_root__polyfun,axiom,
    ! [N2: nat,Z2: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_int @ Z2 @ N2 )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N2 ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_8212_root__polyfun,axiom,
    ! [N2: nat,Z2: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_complex @ Z2 @ N2 )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N2 ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_8213_root__polyfun,axiom,
    ! [N2: nat,Z2: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_rat @ Z2 @ N2 )
          = A )
        = ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I3 = N2 ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_8214_root__polyfun,axiom,
    ! [N2: nat,Z2: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_8256067586552552935nteger @ Z2 @ N2 )
          = A )
        = ( ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I3 = N2 ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_8215_root__polyfun,axiom,
    ! [N2: nat,Z2: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_real @ Z2 @ N2 )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N2 ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_8216_sum__gp0,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_8217_sum__gp0,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_8218_sum__gp0,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_8219_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_8220_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_8221_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_8222_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_8223_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_8224_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8225_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8226_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8227_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8228_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8229_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8230_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( 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 @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8231_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( 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 @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8232_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( 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 @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8233_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( 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 @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8234_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_8235_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_8236_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_8237_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_8238_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_8239_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_8240_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_8241_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > complex,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M8: real] :
        ! [Z4: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8242_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M8: real] :
        ! [Z4: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8243_polyfun__diff,axiom,
    ! [N2: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_complex @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8244_polyfun__diff,axiom,
    ! [N2: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_rat @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8245_polyfun__diff,axiom,
    ! [N2: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_int @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8246_polyfun__diff,axiom,
    ! [N2: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_real @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8247_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N3: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_8248_sin__x__sin__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N3: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_8249_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8250_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8251_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_8252_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z2 )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z2
               != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8253_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_1_of_int_real @ W2 )
        = ( ring_1_of_int_real @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_8254_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_1_of_int_rat @ W2 )
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_8255_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_17405671764205052669omplex @ W2 )
        = ( ring_17405671764205052669omplex @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_8256_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] : N3 ) ) ).

% of_nat_id
thf(fact_8257_scaleR__zero__right,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% scaleR_zero_right
thf(fact_8258_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ( real_V1485227260804924795R_real @ A @ X )
        = ( real_V1485227260804924795R_real @ B @ X ) )
      = ( ( A = B )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
thf(fact_8259_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( ( real_V1485227260804924795R_real @ A @ X )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
thf(fact_8260_scaleR__zero__left,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
      = zero_zero_real ) ).

% scaleR_zero_left
thf(fact_8261_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_8262_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_8263_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_8264_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_8265_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_8266_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_8267_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_8268_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_8269_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = zero_zero_int )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_8270_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = zero_zero_real )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_8271_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = zero_zero_rat )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_8272_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_8273_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = ( numeral_numeral_rat @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8274_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8275_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8276_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8277_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_8278_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_8279_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_8280_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_8281_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_8282_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_8283_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_8284_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
        = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
      = ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_8285_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_8286_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_8287_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_8288_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_8289_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_8290_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_8291_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_8292_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = one_one_int )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_8293_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = one_one_real )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_8294_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = one_one_rat )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_8295_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = one_one_complex )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_8296_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_mult
thf(fact_8297_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_mult
thf(fact_8298_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_mult
thf(fact_8299_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_mult
thf(fact_8300_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_add
thf(fact_8301_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_add
thf(fact_8302_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_add
thf(fact_8303_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_add
thf(fact_8304_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_minus
thf(fact_8305_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_minus
thf(fact_8306_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_minus
thf(fact_8307_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_minus
thf(fact_8308_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_minus
thf(fact_8309_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_diff
thf(fact_8310_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_diff
thf(fact_8311_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_diff
thf(fact_8312_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_diff
thf(fact_8313_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri8010041392384452111omplex @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_8314_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_8315_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_8316_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_8317_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).

% of_int_abs
thf(fact_8318_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
      = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).

% of_int_abs
thf(fact_8319_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
      = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).

% of_int_abs
thf(fact_8320_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
      = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).

% of_int_abs
thf(fact_8321_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_8322_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_8323_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_8324_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_8325_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_8326_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_8327_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_8328_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_8329_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_8330_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_8331_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_8332_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_8333_frac__of__int,axiom,
    ! [Z2: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_8334_frac__of__int,axiom,
    ! [Z2: int] :
      ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = zero_zero_rat ) ).

% frac_of_int
thf(fact_8335_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8336_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8337_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8338_of__int__prod,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A3 ) )
      = ( groups129246275422532515t_real
        @ ^ [X3: nat] : ( ring_1_of_int_real @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8339_of__int__prod,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A3 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X3: nat] : ( ring_1_of_int_rat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8340_of__int__prod,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_17405671764205052669omplex @ ( groups705719431365010083at_int @ F @ A3 ) )
      = ( groups6464643781859351333omplex
        @ ^ [X3: nat] : ( ring_17405671764205052669omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8341_of__int__prod,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A3 ) )
      = ( groups705719431365010083at_int
        @ ^ [X3: nat] : ( ring_1_of_int_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8342_of__int__prod,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A3 ) )
      = ( groups2316167850115554303t_real
        @ ^ [X3: int] : ( ring_1_of_int_real @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8343_of__int__prod,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A3 ) )
      = ( groups1072433553688619179nt_rat
        @ ^ [X3: int] : ( ring_1_of_int_rat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8344_of__int__prod,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_17405671764205052669omplex @ ( groups1705073143266064639nt_int @ F @ A3 ) )
      = ( groups7440179247065528705omplex
        @ ^ [X3: int] : ( ring_17405671764205052669omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8345_of__int__prod,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A3 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : ( ring_1_of_int_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_prod
thf(fact_8346_scaleR__collapse,axiom,
    ! [U: real,A: real] :
      ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_8347_of__int__sum,axiom,
    ! [F: nat > int,A3: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X3: nat] : ( ring_1_of_int_real @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_8348_of__int__sum,axiom,
    ! [F: complex > int,A3: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X3: complex] : ( ring_17405671764205052669omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_8349_of__int__sum,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups8778361861064173332t_real
        @ ^ [X3: int] : ( ring_1_of_int_real @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_8350_of__int__sum,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X3: int] : ( ring_1_of_int_rat @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_8351_of__int__sum,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_17405671764205052669omplex @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups3049146728041665814omplex
        @ ^ [X3: int] : ( ring_17405671764205052669omplex @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_8352_of__int__sum,axiom,
    ! [F: int > int,A3: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( ring_1_of_int_int @ ( F @ X3 ) )
        @ A3 ) ) ).

% of_int_sum
thf(fact_8353_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_8354_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_8355_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_8356_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_8357_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_8358_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_8359_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_8360_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_8361_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_8362_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_8363_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_8364_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_8365_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_8366_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_8367_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_8368_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_8369_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_8370_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_8371_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_8372_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_8373_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_8374_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_8375_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_8376_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_8377_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_8378_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_8379_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_8380_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_8381_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_8382_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_8383_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_8384_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_8385_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_8386_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_8387_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_8388_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_8389_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8390_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8391_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8392_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8393_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8394_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8395_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8396_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8397_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_8398_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_8399_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_8400_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_8401_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_8402_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_8403_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_8404_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_8405_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_8406_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_8407_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_8408_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_8409_sin__npi__int,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8410_scaleR__half__double,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_8411_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_8412_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_8413_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_8414_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_8415_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_8416_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_8417_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_8418_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_8419_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_8420_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_8421_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_8422_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_8423_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8424_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8425_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8426_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8427_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8428_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8429_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8430_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8431_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8432_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8433_sin__int__2pin,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8434_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8435_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8436_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8437_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8438_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8439_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8440_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8441_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8442_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8443_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8444_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8445_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8446_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8447_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8448_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8449_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8450_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ).

% ex_le_of_int
thf(fact_8451_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ).

% ex_le_of_int
thf(fact_8452_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ).

% ex_of_int_less
thf(fact_8453_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ).

% ex_of_int_less
thf(fact_8454_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ).

% ex_less_of_int
thf(fact_8455_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ).

% ex_less_of_int
thf(fact_8456_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_8457_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_8458_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_8459_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_8460_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B: real] :
      ( ( X != zero_zero_real )
     => ( ( ( real_V1485227260804924795R_real @ A @ X )
          = ( real_V1485227260804924795R_real @ B @ X ) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
thf(fact_8461_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).

% scaleR_right_distrib
thf(fact_8462_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_8463_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_8464_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_8465_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_18347121197199848620nteger @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_Code_integer @ ( ring_18347121197199848620nteger @ X ) @ ( ring_18347121197199848620nteger @ Y ) ) ) ).

% of_int_max
thf(fact_8466_Ints__of__int,axiom,
    ! [Z2: int] : ( member_int @ ( ring_1_of_int_int @ Z2 ) @ ring_1_Ints_int ) ).

% Ints_of_int
thf(fact_8467_Ints__of__int,axiom,
    ! [Z2: int] : ( member_real @ ( ring_1_of_int_real @ Z2 ) @ ring_1_Ints_real ) ).

% Ints_of_int
thf(fact_8468_Ints__of__int,axiom,
    ! [Z2: int] : ( member_rat @ ( ring_1_of_int_rat @ Z2 ) @ ring_1_Ints_rat ) ).

% Ints_of_int
thf(fact_8469_Ints__of__int,axiom,
    ! [Z2: int] : ( member_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ ring_1_Ints_complex ) ).

% Ints_of_int
thf(fact_8470_Ints__induct,axiom,
    ! [Q4: int,P: int > $o] :
      ( ( member_int @ Q4 @ ring_1_Ints_int )
     => ( ! [Z: int] : ( P @ ( ring_1_of_int_int @ Z ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_8471_Ints__induct,axiom,
    ! [Q4: real,P: real > $o] :
      ( ( member_real @ Q4 @ ring_1_Ints_real )
     => ( ! [Z: int] : ( P @ ( ring_1_of_int_real @ Z ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_8472_Ints__induct,axiom,
    ! [Q4: rat,P: rat > $o] :
      ( ( member_rat @ Q4 @ ring_1_Ints_rat )
     => ( ! [Z: int] : ( P @ ( ring_1_of_int_rat @ Z ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_8473_Ints__induct,axiom,
    ! [Q4: complex,P: complex > $o] :
      ( ( member_complex @ Q4 @ ring_1_Ints_complex )
     => ( ! [Z: int] : ( P @ ( ring_17405671764205052669omplex @ Z ) )
       => ( P @ Q4 ) ) ) ).

% Ints_induct
thf(fact_8474_Ints__cases,axiom,
    ! [Q4: int] :
      ( ( member_int @ Q4 @ ring_1_Ints_int )
     => ~ ! [Z: int] :
            ( Q4
           != ( ring_1_of_int_int @ Z ) ) ) ).

% Ints_cases
thf(fact_8475_Ints__cases,axiom,
    ! [Q4: real] :
      ( ( member_real @ Q4 @ ring_1_Ints_real )
     => ~ ! [Z: int] :
            ( Q4
           != ( ring_1_of_int_real @ Z ) ) ) ).

% Ints_cases
thf(fact_8476_Ints__cases,axiom,
    ! [Q4: rat] :
      ( ( member_rat @ Q4 @ ring_1_Ints_rat )
     => ~ ! [Z: int] :
            ( Q4
           != ( ring_1_of_int_rat @ Z ) ) ) ).

% Ints_cases
thf(fact_8477_Ints__cases,axiom,
    ! [Q4: complex] :
      ( ( member_complex @ Q4 @ ring_1_Ints_complex )
     => ~ ! [Z: int] :
            ( Q4
           != ( ring_17405671764205052669omplex @ Z ) ) ) ).

% Ints_cases
thf(fact_8478_scale__right__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( nO_MATCH_real_real @ ( divide_divide_real @ X @ Y ) @ A )
     => ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
        = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ) ).

% scale_right_distrib_NO_MATCH
thf(fact_8479_scale__right__distrib__NO__MATCH,axiom,
    ! [X: complex,Y: complex,A: real] :
      ( ( nO_MAT9165723751696580935x_real @ ( divide1717551699836669952omplex @ X @ Y ) @ A )
     => ( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y ) )
        = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ) ).

% scale_right_distrib_NO_MATCH
thf(fact_8480_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).

% scaleR_left_distrib
thf(fact_8481_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa2 )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y @ Xa2 ) ) ) ).

% scaleR_left.add
thf(fact_8482_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( A
          = ( numeral_numeral_real @ W2 ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_8483_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_8484_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_8485_scaleR__right__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).

% scaleR_right_mono
thf(fact_8486_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).

% scaleR_right_mono_neg
thf(fact_8487_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
        = X )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ U @ A )
            = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_8488_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X
        = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ V @ X )
            = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_8489_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_8490_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_8491_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_8492_real__of__int__div4,axiom,
    ! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_8493_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).

% of_int_divide_in_Ints
thf(fact_8494_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).

% of_int_divide_in_Ints
thf(fact_8495_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).

% of_int_divide_in_Ints
thf(fact_8496_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).

% of_int_divide_in_Ints
thf(fact_8497_real__of__int__div,axiom,
    ! [D: int,N2: int] :
      ( ( dvd_dvd_int @ D @ N2 )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_8498_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_8499_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_8500_eval__nat__numeral_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_8501_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_8502_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_8503_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_8504_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_8505_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_8506_of__int__leD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_8507_of__int__leD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_8508_of__int__leD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_8509_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_8510_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_pos
thf(fact_8511_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_pos
thf(fact_8512_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_pos
thf(fact_8513_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
thf(fact_8514_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
thf(fact_8515_of__int__lessD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_8516_of__int__lessD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_8517_of__int__lessD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_8518_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_8519_scaleR__mono,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ X )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).

% scaleR_mono
thf(fact_8520_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).

% scaleR_mono'
thf(fact_8521_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ X @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
     => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).

% split_scaleR_neg_le
thf(fact_8522_split__scaleR__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 @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).

% split_scaleR_pos_le
thf(fact_8523_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).

% scaleR_nonneg_nonneg
thf(fact_8524_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).

% scaleR_nonneg_nonpos
thf(fact_8525_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).

% scaleR_nonpos_nonneg
thf(fact_8526_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).

% scaleR_nonpos_nonpos
thf(fact_8527_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).

% scaleR_left_le_one_le
thf(fact_8528_floor__exists,axiom,
    ! [X: real] :
    ? [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_8529_floor__exists,axiom,
    ! [X: rat] :
    ? [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_8530_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 ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
thf(fact_8531_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 ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
thf(fact_8532_scaleR__2,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
      = ( plus_plus_real @ X @ X ) ) ).

% scaleR_2
thf(fact_8533_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8534_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8535_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8536_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8537_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8538_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8539_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8540_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8541_Complex__sum_H,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [X3: nat] : ( complex2 @ ( F @ X3 ) @ zero_zero_real )
        @ S )
      = ( complex2 @ ( groups6591440286371151544t_real @ F @ S ) @ zero_zero_real ) ) ).

% Complex_sum'
thf(fact_8542_Complex__sum_H,axiom,
    ! [F: complex > real,S: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X3: complex] : ( complex2 @ ( F @ X3 ) @ zero_zero_real )
        @ S )
      = ( complex2 @ ( groups5808333547571424918x_real @ F @ S ) @ zero_zero_real ) ) ).

% Complex_sum'
thf(fact_8543_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N3: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_8544_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N3: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).

% int_less_real_le
thf(fact_8545_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8546_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_8547_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_8548_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N2 ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_8549_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_8550_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bitM @ N2 ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N2 ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_8551_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bitM @ N2 ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_8552_real__of__int__div2,axiom,
    ! [N2: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_8553_real__of__int__div3,axiom,
    ! [N2: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_8554_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8555_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8556_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N3: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_8557_cos__x__cos__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N3: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_8558_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_8559_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_8560_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_8561_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_8562_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8563_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_8564_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_8565_round__unique_H,axiom,
    ! [X: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_8566_round__unique_H,axiom,
    ! [X: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_8567_of__int__round__abs__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_8568_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_8569_exp__first__two__terms,axiom,
    ( exp_real
    = ( ^ [X3: real] :
          ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 )
          @ ( suminf_real
            @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_8570_exp__first__two__terms,axiom,
    ( exp_complex
    = ( ^ [X3: complex] :
          ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X3 )
          @ ( suminf_complex
            @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X3 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_8571_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
      = ( A = B ) ) ).

% inverse_eq_iff_eq
thf(fact_8572_inverse__eq__iff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B ) )
      = ( A = B ) ) ).

% inverse_eq_iff_eq
thf(fact_8573_inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_8574_inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_8575_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_8576_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_8577_inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% inverse_zero
thf(fact_8578_inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% inverse_zero
thf(fact_8579_inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_8580_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_8581_inverse__1,axiom,
    ( ( invers8013647133539491842omplex @ one_one_complex )
    = one_one_complex ) ).

% inverse_1
thf(fact_8582_inverse__1,axiom,
    ( ( inverse_inverse_real @ one_one_real )
    = one_one_real ) ).

% inverse_1
thf(fact_8583_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_8584_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( invers8013647133539491842omplex @ X )
        = one_one_complex )
      = ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
thf(fact_8585_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( inverse_inverse_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% inverse_eq_1_iff
thf(fact_8586_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( inverse_inverse_rat @ X )
        = one_one_rat )
      = ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_8587_inverse__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ B @ A ) ) ).

% inverse_divide
thf(fact_8588_inverse__divide,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ B @ A ) ) ).

% inverse_divide
thf(fact_8589_inverse__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ B @ A ) ) ).

% inverse_divide
thf(fact_8590_inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).

% inverse_minus_eq
thf(fact_8591_inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).

% inverse_minus_eq
thf(fact_8592_inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% inverse_minus_eq
thf(fact_8593_abs__inverse,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).

% abs_inverse
thf(fact_8594_abs__inverse,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
      = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).

% abs_inverse
thf(fact_8595_round__of__int,axiom,
    ! [N2: int] :
      ( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N2 ) )
      = N2 ) ).

% round_of_int
thf(fact_8596_round__of__int,axiom,
    ! [N2: int] :
      ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N2 ) )
      = N2 ) ).

% round_of_int
thf(fact_8597_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_8598_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_8599_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_8600_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_8601_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_8602_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_8603_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% inverse_negative_iff_negative
thf(fact_8604_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% inverse_negative_iff_negative
thf(fact_8605_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_8606_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_8607_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_8608_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_8609_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_8610_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_8611_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_8612_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_8613_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_8614_round__of__nat,axiom,
    ! [N2: nat] :
      ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% round_of_nat
thf(fact_8615_round__of__nat,axiom,
    ! [N2: nat] :
      ( ( archim7778729529865785530nd_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% round_of_nat
thf(fact_8616_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_8617_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_8618_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_8619_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_8620_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
        = one_one_complex ) ) ).

% right_inverse
thf(fact_8621_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_8622_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_8623_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% left_inverse
thf(fact_8624_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_8625_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_8626_inverse__eq__divide__numeral,axiom,
    ! [W2: num] :
      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W2 ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_8627_inverse__eq__divide__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_real @ ( numeral_numeral_real @ W2 ) )
      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W2 ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_8628_inverse__eq__divide__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W2 ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_8629_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_8630_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_8631_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_8632_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_8633_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_8634_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).

% nonzero_norm_inverse
thf(fact_8635_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
        = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).

% nonzero_norm_inverse
thf(fact_8636_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
thf(fact_8637_inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B ) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
thf(fact_8638_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X: real] :
      ( ( ( times_times_real @ Y @ X )
        = ( times_times_real @ X @ Y ) )
     => ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
        = ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_8639_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( ( times_times_rat @ Y @ X )
        = ( times_times_rat @ X @ Y ) )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X )
        = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_8640_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% field_class.field_inverse_zero
thf(fact_8641_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% field_class.field_inverse_zero
thf(fact_8642_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
thf(fact_8643_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
thf(fact_8644_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_8645_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B ) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_8646_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_8647_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_8648_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
       != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_8649_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
       != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_8650_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_real )
       => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
          = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
thf(fact_8651_norm__inverse__le__norm,axiom,
    ! [R2: real,X: real] :
      ( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ R2 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_8652_norm__inverse__le__norm,axiom,
    ! [R2: real,X: complex] :
      ( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ R2 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_8653_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_8654_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_8655_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).

% negative_imp_inverse_negative
thf(fact_8656_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).

% negative_imp_inverse_negative
thf(fact_8657_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_8658_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_8659_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% inverse_negative_imp_negative
thf(fact_8660_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% inverse_negative_imp_negative
thf(fact_8661_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_8662_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_8663_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_8664_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_8665_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_8666_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_8667_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_8668_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_8669_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
          = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_8670_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
          = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_8671_inverse__numeral__1,axiom,
    ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( numera6690914467698888265omplex @ one ) ) ).

% inverse_numeral_1
thf(fact_8672_inverse__numeral__1,axiom,
    ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
    = ( numeral_numeral_real @ one ) ) ).

% inverse_numeral_1
thf(fact_8673_inverse__numeral__1,axiom,
    ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
    = ( numeral_numeral_rat @ one ) ) ).

% inverse_numeral_1
thf(fact_8674_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
        = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_8675_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
        = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_8676_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
        = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_8677_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = one_one_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8678_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = one_one_real )
     => ( ( inverse_inverse_real @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8679_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8680_field__class_Ofield__divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8681_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8682_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8683_divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A2: complex,B2: complex] : ( times_times_complex @ A2 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).

% divide_inverse
thf(fact_8684_divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A2: real,B2: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B2 ) ) ) ) ).

% divide_inverse
thf(fact_8685_divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A2: rat,B2: rat] : ( times_times_rat @ A2 @ ( inverse_inverse_rat @ B2 ) ) ) ) ).

% divide_inverse
thf(fact_8686_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X3: real,Y2: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).

% divide_real_def
thf(fact_8687_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D5: real,E: real] :
          ( ( ord_less_real @ D5 @ E )
         => ( ( P @ D5 )
           => ( P @ E ) ) )
     => ( ! [N: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_8688_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D5: real,E: real] :
          ( ( ord_less_real @ D5 @ E )
         => ( ( P @ D5 )
           => ( P @ E ) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_8689_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N3: nat] :
            ( ( N3 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_8690_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_8691_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_8692_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_8693_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_8694_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_8695_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_8696_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_8697_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_8698_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_8699_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_8700_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_8701_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_8702_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_8703_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_8704_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_8705_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_8706_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_8707_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_8708_cosh__real__nonzero,axiom,
    ! [X: real] :
      ( ( cosh_real @ X )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_8709_sinh__less__cosh__real,axiom,
    ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_less_cosh_real
thf(fact_8710_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_8711_OR__lower,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 @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_8712_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_8713_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_8714_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_8715_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_8716_cosh__real__nonneg__le__iff,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 @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_8717_cosh__real__nonpos__le__iff,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 @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_8718_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_8719_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_8720_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_8721_OR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_8722_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_8723_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_8724_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_8725_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 ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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 ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_8726_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_8727_Suc__0__mod__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N2
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_8728_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_8729_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_8730_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_8731_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_8732_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_8733_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_8734_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_8735_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_8736_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_8737_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_8738_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_8739_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_8740_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_8741_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_8742_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_8743_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_8744_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_8745_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_8746_cot__npi,axiom,
    ! [N2: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_8747_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_8748_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_8749_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_8750_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_8751_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_8752_less__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_8753_log__of__power__eq,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N2 )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_8754_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_8755_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_8756_le__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_8757_log__base__pow,axiom,
    ! [A: real,N2: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N2 ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_8758_log__nat__power,axiom,
    ! [X: real,B: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_8759_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_8760_log__of__power__less,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_8761_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_8762_log__of__power__le,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_8763_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_8764_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_8765_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q6: 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 ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_8766_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_8767_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_8768_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q6: 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 ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_8769_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_8770_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_8771_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_8772_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_8773_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_8774_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_8775_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q6: nat] : ( product_Pair_nat_nat @ ( suc @ Q6 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N3 ) @ N3 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_8776_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8777_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_8778_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_8779_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_8780_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_8781_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_8782_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_8783_floor__eq,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N2 ) ) ) ).

% floor_eq
thf(fact_8784_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_8785_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_8786_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_8787_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_8788_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M6: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N3 ) ) @ M6 ) ) ) ).

% prod_encode_def
thf(fact_8789_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q6: int,R5: int] : ( plus_plus_int @ Q6 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_8790_floor__eq2,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_8791_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_8792_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N3 ) @ ( modulo_modulo_nat @ M6 @ N3 ) ) ) ) ).

% divmod_nat_def
thf(fact_8793_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_8794_floor__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8795_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: 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 @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8796_list__decode_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
             => ( ! [X2: nat,Y4: nat] :
                    ( ( ( product_Pair_nat_nat @ X2 @ Y4 )
                      = ( nat_prod_decode @ N ) )
                   => ( P @ Y4 ) )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_8797_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D6: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z5: int] :
                ( ( ord_less_eq_int @ D6 @ Z6 )
                & ( ord_less_int @ Z6 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_8798_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_8799_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8800_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_8801_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_8802_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_8803_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_8804_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_8805_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_8806_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_8807_powr__numeral,axiom,
    ! [X: real,N2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N2 ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% powr_numeral
thf(fact_8808_powr__non__neg,axiom,
    ! [A: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_8809_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_8810_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_8811_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_8812_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_8813_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_8814_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_8815_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_8816_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_8817_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_8818_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_8819_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_8820_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_8821_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_8822_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_8823_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_8824_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).

% log_base_powr
thf(fact_8825_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_8826_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).

% log_powr
thf(fact_8827_powr__realpow,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
        = ( power_power_real @ X @ N2 ) ) ) ).

% powr_realpow
thf(fact_8828_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_8829_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_8830_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_8831_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_8832_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_8833_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_8834_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_8835_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_8836_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_8837_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_8838_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_8839_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_8840_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_8841_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_8842_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_8843_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_8844_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_8845_powr__neg__numeral,axiom,
    ! [X: real,N2: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8846_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_8847_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D6: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z5: int] :
                ( ( ord_less_eq_int @ D6 @ Z5 )
                & ( ord_less_int @ Z6 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_8848_arctan__def,axiom,
    ( arctan
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
              & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arctan_def
thf(fact_8849_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( ( complex2 @ X @ Y )
        = ( real_V4546457046886955230omplex @ Xa2 ) )
      = ( ( X = Xa2 )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
thf(fact_8850_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X3: real] : ( complex2 @ X3 @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_8851_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_8852_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X3: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_8853_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X3 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_8854_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X3 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_8855_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_8856_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_8857_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_8858_nat__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
      = N2 ) ).

% nat_int
thf(fact_8859_concat__bit__of__zero__2,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_8860_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_8861_nat__of__bool,axiom,
    ! [P: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% nat_of_bool
thf(fact_8862_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_8863_nat__le__0,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ Z2 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_8864_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_8865_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z2 )
        & ( ord_less_int @ W2 @ Z2 ) ) ) ).

% zless_nat_conj
thf(fact_8866_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_8867_nat__zminus__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_8868_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = Z2 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_8869_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_8870_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_8871_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_8872_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_8873_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_8874_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% zero_less_nat_eq
thf(fact_8875_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_8876_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_8877_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_8878_nat__abs__dvd__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
      = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_abs_dvd_iff
thf(fact_8879_dvd__nat__abs__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( dvd_dvd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).

% dvd_nat_abs_iff
thf(fact_8880_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_8881_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% one_less_nat_eq
thf(fact_8882_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_8883_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_8884_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_8885_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_8886_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_8887_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q4 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_8888_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_8889_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_8890_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_8891_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_8892_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_8893_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_8894_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_8895_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% take_bit_nonnegative
thf(fact_8896_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_8897_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_8898_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_8899_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% int_sgnE
thf(fact_8900_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z2 )
            = ( nat2 @ Z7 ) )
          = ( Z2 = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_8901_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).

% all_nat
thf(fact_8902_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
          & ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).

% ex_nat
thf(fact_8903_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_8904_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L ) )
     => ( ( divide_divide_int @ K @ L )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_8905_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W2 @ Z2 ) ) ) ).

% nat_mono_iff
thf(fact_8906_take__bit__decr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_8907_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).

% zless_nat_eq_int_zless
thf(fact_8908_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_8909_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z2 )
      = ( ( M
          = ( nat2 @ Z2 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).

% int_eq_iff
thf(fact_8910_nat__0__le,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = Z2 ) ) ).

% nat_0_le
thf(fact_8911_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_8912_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_8913_int__minus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_8914_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_8915_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_8916_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_8917_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_8918_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_8919_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_8920_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_8921_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_8922_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W2 @ Z2 ) ) ) ).

% nat_less_eq_zless
thf(fact_8923_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).

% nat_le_eq_zle
thf(fact_8924_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ( nat2 @ W2 )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_8925_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_8926_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N3: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ( P @ N3 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_8927_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_8928_nat__add__distrib,axiom,
    ! [Z2: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_8929_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_8930_nat__mult__distrib,axiom,
    ! [Z2: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_8931_Suc__as__int,axiom,
    ( suc
    = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_8932_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_8933_nat__diff__distrib,axiom,
    ! [Z7: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z2 )
       => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_8934_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_8935_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_8936_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_8937_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_8938_nat__power__eq,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( power_power_int @ Z2 @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_8939_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_8940_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_8941_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_8942_floor__eq3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_8943_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_8944_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_8945_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_8946_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_8947_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_8948_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_8949_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_8950_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_8951_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_8952_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( suc @ ( nat2 @ Z2 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_8953_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_8954_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z7: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_8955_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_8956_floor__eq4,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_8957_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_8958_diff__nat__eq__if,axiom,
    ! [Z7: int,Z2: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z2 ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_8959_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_8960_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_8961_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_8962_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_8963_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_8964_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_8965_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_8966_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_8967_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A1 = K3 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q6: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q6 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q6: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q6 @ 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 @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_8968_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q2: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q2 @ A23 ) ) ) )
         => ~ ! [R3: int,Q2: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q2 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_8969_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_8970_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_8971_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_8972_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_8973_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_8974_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_8975_powr__real__of__int,axiom,
    ! [X: real,N2: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
            = ( power_power_real @ X @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_8976_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_8977_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_8978_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_8979_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( 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_unfold
thf(fact_8980_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_8981_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8982_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_8983_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_8984_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_8985_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_8986_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_8987_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_8988_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_8989_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_8990_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_8991_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8992_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_8993_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_8994_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_8995_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% diff_numeral_Suc
thf(fact_8996_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_8997_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% max_numeral_Suc
thf(fact_8998_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_8999_and__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9000_and__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9001_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X4: num] :
            ( ( P @ X4 )
           => ( P @ ( inc @ X4 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_9002_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_9003_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_9004_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_9005_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_9006_AND__upper1_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).

% AND_upper1'
thf(fact_9007_AND__upper2_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).

% AND_upper2'
thf(fact_9008_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_9009_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9010_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_9011_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_9012_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_9013_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_9014_AND__upper1_H_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).

% AND_upper1''
thf(fact_9015_AND__upper2_H_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).

% AND_upper2''
thf(fact_9016_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_9017_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_9018_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_9019_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_9020_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A2: real] : ( if_real @ ( A2 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A2 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9021_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9022_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_9023_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_9024_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9025_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_9026_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_9027_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_9028_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9029_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).

% signed_take_bit_negative_iff
thf(fact_9030_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_9031_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_9032_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9033_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9034_Suc__0__and__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_9035_and__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_9036_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_9037_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_9038_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9039_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% less_mask
thf(fact_9040_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9041_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9042_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_9043_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_9044_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9045_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ) ) ).

% int_bit_bound
thf(fact_9046_Suc__mask__eq__exp,axiom,
    ! [N2: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_mask_eq_exp
thf(fact_9047_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_9048_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9049_mask__half__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_9050_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N3: nat] :
          ( if_nat
          @ ( ( M6 = zero_zero_nat )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9051_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9052_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9053_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( 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 @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( 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 @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9054_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_9055_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_9056_rat__inverse__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_inverse_code
thf(fact_9057_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_9058_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9059_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9060_set__encode__insert,axiom,
    ! [A3: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ N2 @ A3 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A3 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).

% set_encode_insert
thf(fact_9061_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9062_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9063_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_9064_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9065_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9066_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_9067_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M @ N2 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_9068_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_9069_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_9070_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_9071_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9072_quotient__of__denom__pos,axiom,
    ! [R2: rat,P6: int,Q4: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P6 @ Q4 ) )
     => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).

% quotient_of_denom_pos
thf(fact_9073_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_9074_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q6: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C2: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D6: int] : ( ord_less_int @ ( times_times_int @ A2 @ D6 ) @ ( times_times_int @ C2 @ B2 ) )
              @ ( quotient_of @ Q6 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9075_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_9076_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_9077_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9078_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_9079_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z2: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z2 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z2 ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9080_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( 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 ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_9081_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_9082_normalize__negative,axiom,
    ! [Q4: int,P6: int] :
      ( ( ord_less_int @ Q4 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q4 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).

% normalize_negative
thf(fact_9083_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9084_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9085_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9086_normalize__denom__zero,axiom,
    ! [P6: int] :
      ( ( normalize @ ( product_Pair_int_int @ P6 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9087_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_9088_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_9089_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_9090_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_9091_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9092_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9093_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9094_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S2 )
           => ! [T3: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T3 )
               => ( R2
                 != ( plus_plus_rat @ S2 @ T3 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9095_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_rat_def
thf(fact_9096_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_9097_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_9098_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_9099_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_9100_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P6: int,Q4: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P6 @ Q4 ) )
     => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).

% normalize_denom_pos
thf(fact_9101_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P6: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q4 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P6 @ S )
            = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9102_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_9103_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_9104_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_9105_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_9106_arccos__def,axiom,
    ( arccos
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X3 )
              & ( ord_less_eq_real @ X3 @ pi )
              & ( ( cos_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arccos_def
thf(fact_9107_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_9108_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9109_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9110_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_9111_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9112_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9113_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9114_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9115_concat__bit__of__zero__1,axiom,
    ! [N2: nat,L: int] :
      ( ( bit_concat_bit @ N2 @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N2 @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_9116_push__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_Suc_0
thf(fact_9117_XOR__lower,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 @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9118_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M6: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9119_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M6: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9120_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9121_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9122_XOR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_9123_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9124_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9125_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X7: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( ( ord_less_eq_nat @ M9 @ M6 )
         => ! [N3: nat] :
              ( ( ord_less_eq_nat @ M9 @ N3 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9126_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9127_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_9128_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9129_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9130_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9131_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
           => ( P @ M6 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9132_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
            & ( P @ M6 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9133_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9134_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9135_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9136_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_9137_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_9138_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9139_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9140_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9141_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9142_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_Suc_fact
thf(fact_9143_and__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9144_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9145_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9146_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9147_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9148_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_9149_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9150_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_9151_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_9152_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_9153_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_9154_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_9155_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_9156_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9157_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N2 @ M3 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9158_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9159_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9160_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X3: rat] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9161_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X3: real] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
              & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9162_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_9163_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_9164_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9165_Suc__0__mod__numeral,axiom,
    ! [K: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9166_snd__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ M @ N2 ) ) ).

% snd_divmod_nat
thf(fact_9167_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_9168_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X6 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_9169_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_9170_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y6 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y6 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X6 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_9171_Suc__0__div__numeral,axiom,
    ! [K: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_div_numeral
thf(fact_9172_fst__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( divide_divide_nat @ M @ N2 ) ) ).

% fst_divmod_nat
thf(fact_9173_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9174_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y6 )
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y6 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_9175_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_9176_minus__one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9177_one__mod__minus__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9178_numeral__mod__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9179_minus__numeral__mod__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9180_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9181_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_9182_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X3: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y2 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9183_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_9184_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_9185_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9186_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9187_gcd__neg1__int,axiom,
    ! [X: int,Y: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_neg1_int
thf(fact_9188_gcd__neg2__int,axiom,
    ! [X: int,Y: int] :
      ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_neg2_int
thf(fact_9189_gcd__abs2__int,axiom,
    ! [X: int,Y: int] :
      ( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_abs2_int
thf(fact_9190_gcd__abs1__int,axiom,
    ! [X: int,Y: int] :
      ( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_abs1_int
thf(fact_9191_abs__gcd__int,axiom,
    ! [X: int,Y: int] :
      ( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% abs_gcd_int
thf(fact_9192_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
      = ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9193_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N2: num] :
      ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N2 ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_9194_gcd__neg__numeral__1__int,axiom,
    ! [N2: num,X: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X ) ) ).

% gcd_neg_numeral_1_int
thf(fact_9195_gcd__0__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ X @ zero_zero_int )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_int
thf(fact_9196_gcd__0__left__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ zero_zero_int @ X )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_left_int
thf(fact_9197_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( gcd_gcd_int @ X @ Y )
        = ( abs_abs_int @ Y ) ) ) ).

% gcd_proj2_if_dvd_int
thf(fact_9198_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( gcd_gcd_int @ X @ Y )
        = ( abs_abs_int @ X ) ) ) ).

% gcd_proj1_if_dvd_int
thf(fact_9199_gcd__red__int,axiom,
    ( gcd_gcd_int
    = ( ^ [X3: int,Y2: int] : ( gcd_gcd_int @ Y2 @ ( modulo_modulo_int @ X3 @ Y2 ) ) ) ) ).

% gcd_red_int
thf(fact_9200_gcd__idem__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ X @ X )
      = ( abs_abs_int @ X ) ) ).

% gcd_idem_int
thf(fact_9201_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9202_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9203_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N2 ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9204_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_9205_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_9206_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9207_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A )
        & ( dvd_dvd_int @ D @ B )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A @ B ) ) ) ).

% gcd_unique_int
thf(fact_9208_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_9209_gcd__code__int,axiom,
    ( gcd_gcd_int
    = ( ^ [K3: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).

% gcd_code_int
thf(fact_9210_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_9211_root__powr__inverse,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9212_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_9213_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9214_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_9215_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9216_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9217_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9218_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9219_gcd__nat_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( gcd_gcd_nat @ A @ B )
        = A ) ) ).

% gcd_nat.absorb1
thf(fact_9220_gcd__nat_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( gcd_gcd_nat @ A @ B )
        = B ) ) ).

% gcd_nat.absorb2
thf(fact_9221_gcd__nat_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ( dvd_dvd_nat @ A @ C ) ) ) ).

% gcd_nat.bounded_iff
thf(fact_9222_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( gcd_gcd_nat @ X @ Y )
        = X ) ) ).

% gcd_proj1_if_dvd_nat
thf(fact_9223_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( dvd_dvd_nat @ Y @ X )
     => ( ( gcd_gcd_nat @ X @ Y )
        = Y ) ) ).

% gcd_proj2_if_dvd_nat
thf(fact_9224_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9225_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9226_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9227_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9228_real__root__zero,axiom,
    ! [N2: nat] :
      ( ( root @ N2 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9229_gcd__int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_int_int_eq
thf(fact_9230_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9231_real__root__eq__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = ( root @ N2 @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9232_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9233_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9234_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9235_real__root__eq__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9236_real__root__less__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9237_real__root__le__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9238_real__root__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9239_real__root__eq__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9240_gcd__nat__abs__right__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( gcd_gcd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% gcd_nat_abs_right_eq
thf(fact_9241_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
      = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% gcd_nat_abs_left_eq
thf(fact_9242_real__root__lt__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9243_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9244_real__root__le__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9245_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9246_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9247_real__root__lt__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9248_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9249_real__root__le__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9250_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9251_real__root__pow__pos2,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9252_gcd__nat_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ B @ D )
       => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).

% gcd_nat.mono
thf(fact_9253_gcd__nat_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( A
        = ( gcd_gcd_nat @ A @ B ) ) ) ).

% gcd_nat.orderE
thf(fact_9254_gcd__nat_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( gcd_gcd_nat @ A @ B ) )
     => ( dvd_dvd_nat @ A @ B ) ) ).

% gcd_nat.orderI
thf(fact_9255_gcd__nat_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( ( gcd_gcd_nat @ A @ B )
        = A ) ) ).

% gcd_nat.absorb3
thf(fact_9256_gcd__nat_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ( dvd_dvd_nat @ B @ A )
        & ( B != A ) )
     => ( ( gcd_gcd_nat @ A @ B )
        = B ) ) ).

% gcd_nat.absorb4
thf(fact_9257_gcd__nat_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
     => ~ ( ( dvd_dvd_nat @ A @ B )
         => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).

% gcd_nat.boundedE
thf(fact_9258_gcd__nat_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).

% gcd_nat.boundedI
thf(fact_9259_gcd__nat_Oorder__iff,axiom,
    ( dvd_dvd_nat
    = ( ^ [A2: nat,B2: nat] :
          ( A2
          = ( gcd_gcd_nat @ A2 @ B2 ) ) ) ) ).

% gcd_nat.order_iff
thf(fact_9260_gcd__nat_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).

% gcd_nat.cobounded1
thf(fact_9261_gcd__nat_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).

% gcd_nat.cobounded2
thf(fact_9262_gcd__nat_Oabsorb__iff1,axiom,
    ( dvd_dvd_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( gcd_gcd_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% gcd_nat.absorb_iff1
thf(fact_9263_gcd__nat_Oabsorb__iff2,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( gcd_gcd_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% gcd_nat.absorb_iff2
thf(fact_9264_gcd__nat_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).

% gcd_nat.coboundedI1
thf(fact_9265_gcd__nat_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ C )
     => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).

% gcd_nat.coboundedI2
thf(fact_9266_gcd__nat_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
        & ( A
         != ( gcd_gcd_nat @ B @ C ) ) )
     => ~ ( ( ( dvd_dvd_nat @ A @ B )
            & ( A != B ) )
         => ~ ( ( dvd_dvd_nat @ A @ C )
              & ( A != C ) ) ) ) ).

% gcd_nat.strict_boundedE
thf(fact_9267_gcd__nat_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
      = ( ( A
          = ( gcd_gcd_nat @ A @ B ) )
        & ( A != B ) ) ) ).

% gcd_nat.strict_order_iff
thf(fact_9268_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ C )
        & ( A != C ) )
     => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
        & ( ( gcd_gcd_nat @ A @ B )
         != C ) ) ) ).

% gcd_nat.strict_coboundedI1
thf(fact_9269_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ( dvd_dvd_nat @ B @ C )
        & ( B != C ) )
     => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
        & ( ( gcd_gcd_nat @ A @ B )
         != C ) ) ) ).

% gcd_nat.strict_coboundedI2
thf(fact_9270_gcd__unique__nat,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ D @ A )
        & ( dvd_dvd_nat @ D @ B )
        & ! [E3: nat] :
            ( ( ( dvd_dvd_nat @ E3 @ A )
              & ( dvd_dvd_nat @ E3 @ B ) )
           => ( dvd_dvd_nat @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_nat @ A @ B ) ) ) ).

% gcd_unique_nat
thf(fact_9271_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9272_gcd__red__nat,axiom,
    ( gcd_gcd_nat
    = ( ^ [X3: nat,Y2: nat] : ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) ) ).

% gcd_red_nat
thf(fact_9273_real__root__pos__pos__le,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9274_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_9275_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_9276_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_9277_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_9278_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_9279_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X3: nat,Y2: nat] : ( if_nat @ ( Y2 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X3 @ Y2 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9280_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_9281_real__root__less__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9282_real__root__le__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9283_real__root__power,axiom,
    ! [N2: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9284_real__root__abs,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_abs
thf(fact_9285_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_9286_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_9287_sgn__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9288_real__root__gt__zero,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9289_real__root__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9290_root__abs__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9291_gcd__int__def,axiom,
    ( gcd_gcd_int
    = ( ^ [X3: int,Y2: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X3 ) ) @ ( nat2 @ ( abs_abs_int @ Y2 ) ) ) ) ) ) ).

% gcd_int_def
thf(fact_9292_real__root__pos__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9293_real__root__strict__increasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9294_real__root__decreasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9295_real__root__pow__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9296_real__root__pos__unique,axiom,
    ! [N2: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N2 )
            = X )
         => ( ( root @ N2 @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9297_real__root__power__cancel,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9298_real__root__increasing,axiom,
    ! [N2: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9299_root__sgn__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9300_sgn__power__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9301_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9302_ln__root,axiom,
    ! [N2: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N2 @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_9303_log__root,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N2 @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_9304_log__base__root,axiom,
    ! [N2: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9305_split__root,axiom,
    ! [P: real > $o,N2: nat,X: real] :
      ( ( P @ ( root @ N2 @ X ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
                = X )
             => ( P @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_9306_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_9307_gcd__idem__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ X )
      = X ) ).

% gcd_idem_nat
thf(fact_9308_gcd__nat_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
      = ( gcd_gcd_nat @ A @ B ) ) ).

% gcd_nat.right_idem
thf(fact_9309_gcd__nat_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
      = ( gcd_gcd_nat @ A @ B ) ) ).

% gcd_nat.left_idem
thf(fact_9310_gcd__nat_Oidem,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ A )
      = A ) ).

% gcd_nat.idem
thf(fact_9311_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9312_sub__BitM__One__eq,axiom,
    ! [N2: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9313_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
    @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9314_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_9315_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_9316_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9317_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_9318_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_9319_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_9320_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_9321_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_9322_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9323_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9324_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9325_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_9326_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
     => ( A3
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9327_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9328_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_9329_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z5: complex] :
                  ( ( power_power_complex @ Z5 @ N2 )
                  = C ) ) )
          = N2 ) ) ) ).

% card_nth_roots
thf(fact_9330_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) ) )
        = N2 ) ) ).

% card_roots_unity_eq
thf(fact_9331_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M6: nat,N3: nat] :
        ( ( dvd_dvd_nat @ M6 @ N3 )
        & ( M6 != N3 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9332_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_9333_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_9334_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N7: nat] :
              ( ( ord_less_nat @ N7 @ ( finite_card_nat @ S3 ) )
             => ( member_nat @ ( R3 @ N7 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_9335_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_9336_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9337_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9338_Gcd__dvd__nat,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A3 ) @ A ) ) ).

% Gcd_dvd_nat
thf(fact_9339_Gcd__greatest__nat,axiom,
    ! [A3: set_nat,A: nat] :
      ( ! [B4: nat] :
          ( ( member_nat @ B4 @ A3 )
         => ( dvd_dvd_nat @ A @ B4 ) )
     => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A3 ) ) ) ).

% Gcd_greatest_nat
thf(fact_9340_Gcd__nat__eq__one,axiom,
    ! [N6: set_nat] :
      ( ( member_nat @ one_one_nat @ N6 )
     => ( ( gcd_Gcd_nat @ N6 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_9341_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member_nat @ A4 @ A3 )
         => ( ( member_nat @ B4 @ A3 )
           => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A3 ) ) )
     => ( ( A3 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).

% Gcd_in
thf(fact_9342_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_9343_abs__Gcd__eq,axiom,
    ! [K5: set_int] :
      ( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
      = ( gcd_Gcd_int @ K5 ) ) ).

% abs_Gcd_eq
thf(fact_9344_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9345_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_9346_Gcd__dvd__int,axiom,
    ! [A: int,A3: set_int] :
      ( ( member_int @ A @ A3 )
     => ( dvd_dvd_int @ ( gcd_Gcd_int @ A3 ) @ A ) ) ).

% Gcd_dvd_int
thf(fact_9347_Gcd__greatest__int,axiom,
    ! [A3: set_int,A: int] :
      ( ! [B4: int] :
          ( ( member_int @ B4 @ A3 )
         => ( dvd_dvd_int @ A @ B4 ) )
     => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A3 ) ) ) ).

% Gcd_greatest_int
thf(fact_9348_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9349_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_9350_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
          @ ^ [X3: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9351_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X4: nat,Y3: nat] :
          ( Z2
         != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_9352_int_Oabs__induct,axiom,
    ! [P: int > $o,X: int] :
      ( ! [Y3: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y3 ) )
     => ( P @ X ) ) ).

% int.abs_induct
thf(fact_9353_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9354_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9355_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_9356_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_9357_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9358_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
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9359_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
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9360_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
          @ ^ [X3: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9361_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
          @ ^ [X3: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9362_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9363_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9364_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9365_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q4 ) )
      = Q4 ) ).

% num_of_nat_numeral_eq
thf(fact_9366_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9367_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_9368_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_9369_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).

% nat.rep_eq
thf(fact_9370_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_9371_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9372_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) ) ) ) ).

% uminus_int_def
thf(fact_9373_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_9374_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9375_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9376_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% add_rat
thf(fact_9377_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9378_eq__rat_I3_J,axiom,
    ! [A: int,C: int] :
      ( ( fract @ zero_zero_int @ A )
      = ( fract @ zero_zero_int @ C ) ) ).

% eq_rat(3)
thf(fact_9379_eq__rat_I2_J,axiom,
    ! [A: int] :
      ( ( fract @ A @ zero_zero_int )
      = ( fract @ zero_zero_int @ one_one_int ) ) ).

% eq_rat(2)
thf(fact_9380_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( P @ ( fract @ A4 @ B4 ) ) )
     => ( P @ Q4 ) ) ).

% Rat_induct_pos
thf(fact_9381_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9382_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( fract @ A @ B ) ) ) ).

% mult_rat_cancel
thf(fact_9383_rat__number__collapse_I6_J,axiom,
    ! [K: int] :
      ( ( fract @ K @ zero_zero_int )
      = zero_zero_rat ) ).

% rat_number_collapse(6)
thf(fact_9384_rat__number__collapse_I1_J,axiom,
    ! [K: int] :
      ( ( fract @ zero_zero_int @ K )
      = zero_zero_rat ) ).

% rat_number_collapse(1)
thf(fact_9385_Zero__rat__def,axiom,
    ( zero_zero_rat
    = ( fract @ zero_zero_int @ one_one_int ) ) ).

% Zero_rat_def
thf(fact_9386_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_9387_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_9388_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_9389_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_9390_Fract__add__one,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M @ N2 ) @ N2 )
        = ( plus_plus_rat @ ( fract @ M @ N2 ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_9391_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_9392_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_9393_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_9394_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_9395_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9396_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9397_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9398_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9399_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q6: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q6 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9400_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9401_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9402_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9403_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9404_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9405_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_9406_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ) ).

% less_int_def
thf(fact_9407_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
        @ ^ [X3: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9408_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_9409_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_9410_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_9411_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9412_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9413_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9414_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9415_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M6: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9416_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9417_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9418_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9419_Gcd__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
      = ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_abs_eq
thf(fact_9420_Gcd__int__eq,axiom,
    ! [N6: set_nat] :
      ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N6 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N6 ) ) ) ).

% Gcd_int_eq
thf(fact_9421_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_nat
        @ ( image_int_nat
          @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
          @ K5 ) )
      = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).

% Gcd_nat_abs_eq
thf(fact_9422_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9423_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q6: num] : ( some_num @ ( bit0 @ Q6 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9424_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9425_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_9426_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_9427_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9428_and__not__num__eq__None__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9429_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_9430_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9431_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_9432_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_9433_Rat_Opositive__zero,axiom,
    ~ ( positive @ zero_zero_rat ) ).

% Rat.positive_zero
thf(fact_9434_Rat_Opositive__minus,axiom,
    ! [X: rat] :
      ( ~ ( positive @ X )
     => ( ( X != zero_zero_rat )
       => ( positive @ ( uminus_uminus_rat @ X ) ) ) ) ).

% Rat.positive_minus
thf(fact_9435_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] : ( positive @ ( minus_minus_rat @ Y2 @ X3 ) ) ) ) ).

% less_rat_def
thf(fact_9436_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X3: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X3 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X3 ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9437_Gcd__int__def,axiom,
    ( gcd_Gcd_int
    = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).

% Gcd_int_def
thf(fact_9438_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).

% Rat.positive_def
thf(fact_9439_measure__function__int,axiom,
    fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).

% measure_function_int
thf(fact_9440_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_9441_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9442_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9443_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_9444_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D6: nat] : ( dvd_dvd_nat @ D6 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_9445_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9446_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D6: int] : ( dvd_dvd_int @ D6 @ N2 ) ) )
        = ( abs_abs_int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_9447_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_9448_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9449_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N2 )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D6: int] :
                ( ( dvd_dvd_int @ D6 @ M )
                & ( dvd_dvd_int @ D6 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_9450_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9451_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N3 ) @ M6 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9452_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D6: nat] :
                ( ( dvd_dvd_nat @ D6 @ M )
                & ( dvd_dvd_nat @ D6 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9453_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_9454_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9455_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_9456_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_9457_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_9458_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9459_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_9460_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9461_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_9462_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M6: nat] :
                      ( collect_nat
                      @ ^ [D6: nat] : ( dvd_dvd_nat @ D6 @ M6 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9463_min__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q4 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_9464_min__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q4 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_9465_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_9466_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9467_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9468_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% min_numeral_Suc
thf(fact_9469_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9470_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).

% min_diff
thf(fact_9471_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q4 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).

% nat_mult_min_left
thf(fact_9472_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q4 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).

% nat_mult_min_right
thf(fact_9473_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M3: nat] : ( suc @ ( ord_min_nat @ M3 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_9474_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M3: nat] : ( suc @ ( ord_min_nat @ N2 @ M3 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_9475_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9476_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X7: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X7 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9477_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_9478_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_9479_suminf__eq__SUP__real,axiom,
    ! [X6: nat > real] :
      ( ( summable_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X6 @ I2 ) )
       => ( ( suminf_real @ X6 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X6 @ ( set_ord_lessThan_nat @ I3 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9480_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).

% Sup_nat_def
thf(fact_9481_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9482_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_9483_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9484_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_9485_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_9486_surj__prod__decode,axiom,
    ( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% surj_prod_decode
thf(fact_9487_surj__prod__encode,axiom,
    ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% surj_prod_encode
thf(fact_9488_int__in__range__abs,axiom,
    ! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).

% int_in_range_abs
thf(fact_9489_root__def,axiom,
    ( root
    = ( ^ [N3: nat,X3: real] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N3 ) )
            @ X3 ) ) ) ) ).

% root_def
thf(fact_9490_DERIV__real__root__generic,axiom,
    ! [N2: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9491_DERIV__even__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9492_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F5: 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 @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
       => ? [Z: real] :
            ( ( ord_less_real @ A @ Z )
            & ( ord_less_real @ Z @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z ) ) ) ) ) ) ).

% MVT2
thf(fact_9493_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9494_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9495_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9496_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 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9497_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 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9498_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_9499_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9500_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 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D5 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9501_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 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D5 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9502_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_9503_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_9504_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 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D5 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9505_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 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D5 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9506_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D5 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9507_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D5 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9508_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D5 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9509_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D5 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9510_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_9511_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_9512_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_9513_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N2: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N2 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9514_DERIV__pow,axiom,
    ! [N2: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X3: real] : ( power_power_real @ X3 @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_9515_has__real__derivative__powr,axiom,
    ! [Z2: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z5: real] : ( powr_real @ Z5 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9516_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_9517_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9518_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( 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
            @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
            @ ( 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 @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9519_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_9520_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9521_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
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X4 @ N3 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X3: real] :
                ( suminf_real
                @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X3 @ ( suc @ N3 ) ) ) )
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9522_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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 @ A3 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9523_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: 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 @ A3 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9524_DERIV__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9525_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_9526_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_9527_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9528_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9529_DERIV__odd__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9530_Maclaurin,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M5: nat,T3: real] :
                ( ( ( ord_less_nat @ M5 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
                  & ( ord_less_eq_real @ T3 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ T3 )
                & ( ord_less_real @ T3 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9531_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T3 )
                & ( ord_less_eq_real @ T3 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ? [T3: real] :
              ( ( ord_less_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9532_Maclaurin__minus,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M5: nat,T3: real] :
                ( ( ( ord_less_nat @ M5 @ N2 )
                  & ( ord_less_eq_real @ H2 @ T3 )
                  & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ H2 @ T3 )
                & ( ord_less_real @ T3 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9533_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X != zero_zero_real )
         => ( ! [M5: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
                & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9534_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M5: nat,T3: real] :
            ( ( ( ord_less_nat @ M5 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9535_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T3: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T3 )
                            & ( ord_less_real @ T3 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T3 )
                            & ( ord_less_real @ T3 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M6 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9536_Taylor__up,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T3: real] :
                  ( ( ord_less_real @ C @ T3 )
                  & ( ord_less_real @ T3 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9537_Taylor__down,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T3: real] :
                  ( ( ord_less_real @ A @ T3 )
                  & ( ord_less_real @ T3 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9538_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
      ( ! [M5: nat,T3: real] :
          ( ( ( ord_less_nat @ M5 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M2: nat,T6: real] :
            ( ( ( ord_less_nat @ M2 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M2 ) ) )
                    @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T6 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T6 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9539_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: 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 @ X9 @ ( 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_9540_surj__int__encode,axiom,
    ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
    = top_top_set_nat ) ).

% surj_int_encode
thf(fact_9541_int__encode__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( nat_int_encode @ X )
        = ( nat_int_encode @ Y ) )
      = ( X = Y ) ) ).

% int_encode_eq
thf(fact_9542_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9543_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
               => ( ( F @ X2 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9544_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9545_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 )
       => ( ! [Z: real] :
              ( ( ord_less_eq_real @ A @ Z )
             => ( ( ord_less_eq_real @ Z @ B )
               => ( ( G @ ( F @ Z ) )
                  = Z ) ) )
         => ( ! [Z: real] :
                ( ( ord_less_eq_real @ A @ Z )
               => ( ( ord_less_eq_real @ Z @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9546_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).

% isCont_ln
thf(fact_9547_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_9548_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_9549_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D4 != 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 @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9550_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_9551_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_9552_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_9553_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_9554_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
           => ( ( G @ ( F @ Z ) )
              = Z ) )
       => ( ! [Z: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9555_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z: real] :
            ( ( ord_less_eq_real @ A @ Z )
           => ( ( ord_less_eq_real @ Z @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ F ) ) )
       => ( ! [Z: real] :
              ( ( ord_less_eq_real @ A @ Z )
             => ( ( ord_less_eq_real @ Z @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ G ) ) )
         => ( ! [Z: real] :
                ( ( ord_less_real @ A @ Z )
               => ( ( ord_less_real @ Z @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z ) @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) )
           => ( ! [Z: real] :
                  ( ( ord_less_real @ A @ Z )
                 => ( ( ord_less_real @ Z @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z ) @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9556_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 )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9557_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 ) )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9558_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9559_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9560_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_9561_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N ) ) @ ( G @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
         => ( ( filterlim_nat_real
              @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N7: nat] : ( ord_less_eq_real @ L4 @ ( G @ N7 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9562_LIMSEQ__inverse__zero,axiom,
    ! [X6: nat > real] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N8 @ N )
         => ( ord_less_real @ R3 @ ( X6 @ N ) ) )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( X6 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9563_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9564_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( root @ N3 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9565_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9566_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9567_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N7: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N7 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9568_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_9569_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9570_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9571_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9572_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9573_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9574_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9575_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9576_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).

% summable
thf(fact_9577_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9578_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9579_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9580_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9581_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9582_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N7: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N7: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9583_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
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9584_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9585_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9586_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ one_one_real @ Y2 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_9587_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).

% tendsto_arcosh_at_left_1
thf(fact_9588_filterlim__pow__at__bot__odd,axiom,
    ! [N2: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N2 )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9589_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9590_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9591_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9592_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9593_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N5: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ N5 @ N3 )
           => ( P @ N3 ) ) ) ) ).

% eventually_sequentially
thf(fact_9594_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( ord_less_eq_nat @ C @ X4 )
         => ( P @ X4 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9595_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
      = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F3 ) ) ) ).

% le_sequentially
thf(fact_9596_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_9597_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( eventually_real
        @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_9598_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9599_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_9600_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( eventually_real
        @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_9601_exp__at__bot,axiom,
    filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).

% exp_at_bot
thf(fact_9602_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_9603_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_bot_neg
thf(fact_9604_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_9605_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_9606_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_9607_ln__at__0,axiom,
    filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% ln_at_0
thf(fact_9608_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_9609_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_9610_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_9611_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G0 @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_9612_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ X4 @ B )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9613_eventually__at__right__to__top,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_9614_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_9615_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_9616_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_9617_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_top_right
thf(fact_9618_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).

% filterlim_inverse_at_right_top
thf(fact_9619_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_9620_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_9621_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_9622_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_9623_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ B @ X4 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ 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_9624_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_9625_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_9626_filterlim__pow__at__bot__even,axiom,
    ! [N2: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N2 )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9627_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9628_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9629_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9630_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ( 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 @ TreeList )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                  @ ( 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 ) )
                        & ! [I3: nat] :
                            ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                            & ! [X3: nat] :
                                ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                                 => ( ( ord_less_nat @ Mi3 @ X3 )
                                    & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9631_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ( vEBT_VEBT_valid @ X2 @ ( 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 @ TreeList )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                    @ ( 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 ) )
                          & ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X3: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                              & ! [X3: nat] :
                                  ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                                   => ( ( ord_less_nat @ Mi3 @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9632_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X3 @ ( 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 @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
            & ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
          @ ( 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 ) )
                & ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X3: nat] :
                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9633_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( 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 @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                        @ ( 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 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9634_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ( 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 @ TreeList )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                        & ! [X3: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                           => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                      @ ( 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 ) )
                            & ! [I3: nat] :
                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X3: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                & ! [X3: nat] :
                                    ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                                     => ( ( ord_less_nat @ Mi3 @ X3 )
                                        & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9635_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_VEBT_valid @ X2 @ ( 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 @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                        @ ( 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 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9636_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,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( 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 @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                        @ ( 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 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9637_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,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ 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_9638_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,Z: real] :
              ( ( ord_less_real @ A @ Z )
              & ( ord_less_real @ Z @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_9639_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F5: 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 @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
           => ? [Z: real] :
                ( ( ord_less_real @ A @ Z )
                & ( ord_less_real @ Z @ B )
                & ( ( F5 @ Z )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_9640_mvt,axiom,
    ! [A: real,B: real,F: real > real,F5: 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 @ ( F5 @ 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 ) )
                   != ( F5 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_9641_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 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_9642_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 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ 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_9643_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_9644_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_9645_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 ) ) ) )
           => ? [Z: real] :
                ( ( ord_less_real @ A @ Z )
                & ( ord_less_real @ Z @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_9646_range__abs__Nats,axiom,
    ( ( image_int_int @ abs_abs_int @ top_top_set_int )
    = semiring_1_Nats_int ) ).

% range_abs_Nats
thf(fact_9647_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_9648_mono__times__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).

% mono_times_nat
thf(fact_9649_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_9650_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9651_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_9652_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,N3: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
          & ( N3 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9653_surj__int__decode,axiom,
    ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
    = top_top_set_int ) ).

% surj_int_decode
thf(fact_9654_Rats__abs__iff,axiom,
    ! [X: real] :
      ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
      = ( member_real @ X @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_9655_int__decode__inverse,axiom,
    ! [N2: nat] :
      ( ( nat_int_encode @ ( nat_int_decode @ N2 ) )
      = N2 ) ).

% int_decode_inverse
thf(fact_9656_int__encode__inverse,axiom,
    ! [X: int] :
      ( ( nat_int_decode @ ( nat_int_encode @ X ) )
      = X ) ).

% int_encode_inverse
thf(fact_9657_inj__on__diff__nat,axiom,
    ! [N6: set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ N6 )
         => ( ord_less_eq_nat @ K @ N ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
        @ N6 ) ) ).

% inj_on_diff_nat
thf(fact_9658_inj__prod__encode,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A3 ) ).

% inj_prod_encode
thf(fact_9659_inj__prod__decode,axiom,
    ! [A3: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A3 ) ).

% inj_prod_decode
thf(fact_9660_inj__Suc,axiom,
    ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).

% inj_Suc
thf(fact_9661_inj__int__decode,axiom,
    ! [A3: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A3 ) ).

% inj_int_decode
thf(fact_9662_inj__int__encode,axiom,
    ! [A3: set_int] : ( inj_on_int_nat @ nat_int_encode @ A3 ) ).

% inj_int_encode
thf(fact_9663_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( ( member_real @ X4 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X @ X4 ) ) ).

% Rats_no_top_le
thf(fact_9664_int__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_int_decode @ X )
        = ( nat_int_decode @ Y ) )
      = ( X = Y ) ) ).

% int_decode_eq
thf(fact_9665_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_9666_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_9667_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_9668_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_9669_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_9670_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,J3: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_9671_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
         => ( ! [X4: nat] :
                ( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X4 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_9672_powr__real__of__int_H,axiom,
    ! [X: real,N2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( X != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N2 ) )
       => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
          = ( power_int_real @ X @ N2 ) ) ) ) ).

% powr_real_of_int'
thf(fact_9673_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M6: nat,N3: nat] :
            ( N3
            = ( suc @ M6 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9674_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_9675_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_9676_sqr_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N2 ) ) ) ) ).

% sqr.simps(2)
thf(fact_9677_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9678_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).

% sqr_conv_mult
thf(fact_9679_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9680_sqr_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).

% sqr.simps(3)
thf(fact_9681_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F5: real > real] :
      ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X4 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_9682_Arg__bounded,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
      & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).

% Arg_bounded
thf(fact_9683_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( 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 @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9684_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9685_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).

% strict_mono_imp_increasing
thf(fact_9686_cis__Arg,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( cis @ ( arg @ Z2 ) )
        = ( sgn_sgn_complex @ Z2 ) ) ) ).

% cis_Arg
thf(fact_9687_cis__neq__zero,axiom,
    ! [A: real] :
      ( ( cis @ A )
     != zero_zero_complex ) ).

% cis_neq_zero
thf(fact_9688_cis__Arg__unique,axiom,
    ! [Z2: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z2 )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z2 )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_9689_Arg__correct,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z2 )
          = ( cis @ ( arg @ Z2 ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9690_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9691_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G )
         => ( ( bfun_nat_real
              @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_9692_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9693_Arg__def,axiom,
    ( arg
    = ( ^ [Z5: complex] :
          ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A2: real] :
                ( ( ( sgn_sgn_complex @ Z5 )
                  = ( cis @ A2 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A2 )
                & ( ord_less_eq_real @ A2 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9694_bij__betw__Suc,axiom,
    ! [M7: set_nat,N6: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N6 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N6 ) ) ).

% bij_betw_Suc
thf(fact_9695_bij__int__decode,axiom,
    bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).

% bij_int_decode
thf(fact_9696_bij__int__encode,axiom,
    bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).

% bij_int_encode
thf(fact_9697_bij__prod__decode,axiom,
    bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).

% bij_prod_decode
thf(fact_9698_bij__prod__encode,axiom,
    bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).

% bij_prod_encode
thf(fact_9699_quotient__of__def,axiom,
    ( quotient_of
    = ( ^ [X3: rat] :
          ( the_Pr4378521158711661632nt_int
          @ ^ [Pair: product_prod_int_int] :
              ( ( X3
                = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).

% quotient_of_def
thf(fact_9700_normalize__stable,axiom,
    ! [Q4: int,P6: int] :
      ( ( ord_less_int @ zero_zero_int @ Q4 )
     => ( ( algebr932160517623751201me_int @ P6 @ Q4 )
       => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q4 ) )
          = ( product_Pair_int_int @ P6 @ Q4 ) ) ) ) ).

% normalize_stable
thf(fact_9701_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( ( algebr932160517623751201me_int @ A @ D )
     => ( ( algebr932160517623751201me_int @ B @ C )
       => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
            = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
          = ( ( ( abs_abs_int @ A )
              = ( abs_abs_int @ B ) )
            & ( ( abs_abs_int @ C )
              = ( abs_abs_int @ D ) ) ) ) ) ) ).

% coprime_crossproduct_int
thf(fact_9702_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q4
            = ( fract @ A4 @ B4 ) )
         => ( ( ord_less_int @ zero_zero_int @ B4 )
           => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) ).

% Rat_cases
thf(fact_9703_Rat__induct,axiom,
    ! [P: rat > $o,Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ( algebr932160517623751201me_int @ A4 @ B4 )
           => ( P @ ( fract @ A4 @ B4 ) ) ) )
     => ( P @ Q4 ) ) ).

% Rat_induct
thf(fact_9704_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q4
            = ( fract @ A4 @ B4 ) )
         => ( ( ord_less_int @ zero_zero_int @ B4 )
           => ( ( A4 != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A4 @ B4 ) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_9705_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X4: product_prod_int_int] :
      ( ( R2
        = ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
      & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
      & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2
              = ( fract @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ Y4 ) ) )
            & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y4 ) )
            & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ Y4 ) ) )
         => ( Y4 = X4 ) ) ) ).

% quotient_of_unique
thf(fact_9706_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( ( algebr934650988132801477me_nat @ A @ D )
     => ( ( algebr934650988132801477me_nat @ B @ C )
       => ( ( ( times_times_nat @ A @ C )
            = ( times_times_nat @ B @ D ) )
          = ( ( A = B )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
thf(fact_9707_coprime__Suc__0__left,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N2 ) ).

% coprime_Suc_0_left
thf(fact_9708_coprime__Suc__0__right,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_9709_coprime__Suc__right__nat,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ N2 ) ) ).

% coprime_Suc_right_nat
thf(fact_9710_coprime__Suc__left__nat,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ N2 ) @ N2 ) ).

% coprime_Suc_left_nat
thf(fact_9711_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( algebr934650988132801477me_nat @ A @ B )
     => ( ( dvd_dvd_nat @ X @ A )
       => ( ( dvd_dvd_nat @ X @ B )
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
thf(fact_9712_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ N2 ) ) ).

% coprime_diff_one_left_nat
thf(fact_9713_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_9714_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( ( member_real @ X @ field_5140801741446780682s_real )
     => ~ ! [M5: nat,N: nat] :
            ( ( N != zero_zero_nat )
           => ( ( ( abs_abs_real @ X )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri5074537144036343181t_real @ N ) ) )
             => ~ ( algebr934650988132801477me_nat @ M5 @ N ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_9715_Gcd__nat__set__eq__fold,axiom,
    ! [Xs2: list_nat] :
      ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
      = ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).

% Gcd_nat_set_eq_fold
thf(fact_9716_Gcd__int__set__eq__fold,axiom,
    ! [Xs2: list_int] :
      ( ( gcd_Gcd_int @ ( set_int2 @ Xs2 ) )
      = ( fold_int_int @ gcd_gcd_int @ Xs2 @ zero_zero_int ) ) ).

% Gcd_int_set_eq_fold
thf(fact_9717_card__length__sum__list__rec,axiom,
    ! [M: nat,N6: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M )
                & ( ( 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 @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N6 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N6 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9718_card__length__sum__list,axiom,
    ! [M: nat,N6: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N6 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M ) @ one_one_nat ) @ N6 ) ) ).

% card_length_sum_list
thf(fact_9719_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ M ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_9720_take__upt,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
     => ( ( take_nat @ M @ ( upt @ I @ N2 ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_9721_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_9722_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).

% sum_list_upt
thf(fact_9723_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_9724_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_9725_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_9726_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M6: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M6 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9727_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast_upt
thf(fact_9728_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M6 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_9729_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M6 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_9730_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_9731_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_9732_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_9733_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_9734_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q4: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M @ Q4 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9735_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_9736_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9737_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9738_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9739_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9740_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_9741_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_9742_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_9743_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9744_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X4: nat,Xs3: list_nat] :
            ( X
           != ( cons_nat @ X4 @ Xs3 ) ) ) ).

% list_encode.cases
thf(fact_9745_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_9746_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_9747_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_9748_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9749_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9750_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_9751_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ N ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X3: nat,Y2: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y2 ) )
                      @ ( nat_prod_decode @ N ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_9752_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).

% bij_list_decode
thf(fact_9753_list__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_list_decode @ X )
        = ( nat_list_decode @ Y ) )
      = ( X = Y ) ) ).

% list_decode_eq
thf(fact_9754_inj__list__decode,axiom,
    ! [A3: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A3 ) ).

% inj_list_decode
thf(fact_9755_surj__list__decode,axiom,
    ( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
    = top_top_set_list_nat ) ).

% surj_list_decode
thf(fact_9756_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_9757_list__decode_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( nat_list_decode @ ( suc @ N2 ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X3: nat,Y2: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y2 ) )
        @ ( nat_prod_decode @ N2 ) ) ) ).

% list_decode.simps(2)
thf(fact_9758_list__decode_Opsimps_I1_J,axiom,
    ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
   => ( ( nat_list_decode @ zero_zero_nat )
      = nil_nat ) ) ).

% list_decode.psimps(1)
thf(fact_9759_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N: nat] :
              ( ( X
                = ( suc @ N ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X3: nat,Y2: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y2 ) )
                  @ ( nat_prod_decode @ N ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_9760_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
     => ( ( nat_list_decode @ ( suc @ N2 ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X3: nat,Y2: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y2 ) )
          @ ( nat_prod_decode @ N2 ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_9761_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs3: list_nat] :
              ( ( X
                = ( cons_nat @ X4 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_9762_list__encode__inverse,axiom,
    ! [X: list_nat] :
      ( ( nat_list_decode @ ( nat_list_encode @ X ) )
      = X ) ).

% list_encode_inverse
thf(fact_9763_list__decode__inverse,axiom,
    ! [N2: nat] :
      ( ( nat_list_encode @ ( nat_list_decode @ N2 ) )
      = N2 ) ).

% list_decode_inverse
thf(fact_9764_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_9765_inj__list__encode,axiom,
    ! [A3: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A3 ) ).

% inj_list_encode
thf(fact_9766_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_9767_bij__list__encode,axiom,
    bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).

% bij_list_encode
thf(fact_9768_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_9769_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_9770_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9771_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9772_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_9773_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,Xs3: list_nat] :
                ( ( X
                  = ( cons_nat @ X4 @ Xs3 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_9774_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
thf(fact_9775_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9776_cauchy__def,axiom,
    ( cauchy
    = ( ^ [X7: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ K3 @ M6 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M6 ) @ ( X7 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).

% cauchy_def
thf(fact_9777_cauchy__minus,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( cauchy
        @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X6 @ N3 ) ) ) ) ).

% cauchy_minus
thf(fact_9778_cauchy__const,axiom,
    ! [X: rat] :
      ( cauchy
      @ ^ [N3: nat] : X ) ).

% cauchy_const
thf(fact_9779_cauchy__add,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( cauchy
          @ ^ [N3: nat] : ( plus_plus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ).

% cauchy_add
thf(fact_9780_cauchy__diff,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( cauchy
          @ ^ [N3: nat] : ( minus_minus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ).

% cauchy_diff
thf(fact_9781_cauchy__mult,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( cauchy
          @ ^ [N3: nat] : ( times_times_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ).

% cauchy_mult
thf(fact_9782_cauchy__imp__bounded,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ? [B4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ B4 )
          & ! [N7: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N7 ) ) @ B4 ) ) ) ).

% cauchy_imp_bounded
thf(fact_9783_cauchyD,axiom,
    ! [X6: nat > rat,R2: rat] :
      ( ( cauchy @ X6 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [M2: nat] :
            ( ( ord_less_eq_nat @ K2 @ M2 )
           => ! [N7: nat] :
                ( ( ord_less_eq_nat @ K2 @ N7 )
               => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M2 ) @ ( X6 @ N7 ) ) ) @ R2 ) ) ) ) ) ).

% cauchyD
thf(fact_9784_cauchyI,axiom,
    ! [X6: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ K4 @ M5 )
             => ! [N: nat] :
                  ( ( ord_less_eq_nat @ K4 @ N )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M5 ) @ ( X6 @ N ) ) ) @ R3 ) ) ) )
     => ( cauchy @ X6 ) ) ).

% cauchyI
thf(fact_9785_le__Real,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( ( ord_less_eq_real @ ( real2 @ X6 ) @ ( real2 @ Y6 ) )
          = ( ! [R5: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ R5 )
               => ? [K3: nat] :
                  ! [N3: nat] :
                    ( ( ord_less_eq_nat @ K3 @ N3 )
                   => ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( plus_plus_rat @ ( Y6 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).

% le_Real
thf(fact_9786_cauchy__not__vanishes,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ? [B4: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B4 )
            & ? [K2: nat] :
              ! [N7: nat] :
                ( ( ord_less_eq_nat @ K2 @ N7 )
               => ( ord_less_rat @ B4 @ ( abs_abs_rat @ ( X6 @ N7 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes
thf(fact_9787_vanishes__const,axiom,
    ! [C: rat] :
      ( ( vanishes
        @ ^ [N3: nat] : C )
      = ( C = zero_zero_rat ) ) ).

% vanishes_const
thf(fact_9788_eq__Real,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( ( ( real2 @ X6 )
            = ( real2 @ Y6 ) )
          = ( vanishes
            @ ^ [N3: nat] : ( minus_minus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ) ).

% eq_Real
thf(fact_9789_inverse__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( ( vanishes @ X6 )
         => ( ( inverse_inverse_real @ ( real2 @ X6 ) )
            = zero_zero_real ) )
        & ( ~ ( vanishes @ X6 )
         => ( ( inverse_inverse_real @ ( real2 @ X6 ) )
            = ( real2
              @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X6 @ N3 ) ) ) ) ) ) ) ).

% inverse_Real
thf(fact_9790_Real__induct,axiom,
    ! [P: real > $o,X: real] :
      ( ! [X10: nat > rat] :
          ( ( cauchy @ X10 )
         => ( P @ ( real2 @ X10 ) ) )
     => ( P @ X ) ) ).

% Real_induct
thf(fact_9791_mult__Real,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( ( times_times_real @ ( real2 @ X6 ) @ ( real2 @ Y6 ) )
          = ( real2
            @ ^ [N3: nat] : ( times_times_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ) ).

% mult_Real
thf(fact_9792_vanishes__add,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( vanishes @ X6 )
     => ( ( vanishes @ Y6 )
       => ( vanishes
          @ ^ [N3: nat] : ( plus_plus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ).

% vanishes_add
thf(fact_9793_vanishes__minus,axiom,
    ! [X6: nat > rat] :
      ( ( vanishes @ X6 )
     => ( vanishes
        @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X6 @ N3 ) ) ) ) ).

% vanishes_minus
thf(fact_9794_of__nat__Real,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [X3: nat] :
          ( real2
          @ ^ [N3: nat] : ( semiri681578069525770553at_rat @ X3 ) ) ) ) ).

% of_nat_Real
thf(fact_9795_vanishes__diff,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( vanishes @ X6 )
     => ( ( vanishes @ Y6 )
       => ( vanishes
          @ ^ [N3: nat] : ( minus_minus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ).

% vanishes_diff
thf(fact_9796_zero__real__def,axiom,
    ( zero_zero_real
    = ( real2
      @ ^ [N3: nat] : zero_zero_rat ) ) ).

% zero_real_def
thf(fact_9797_one__real__def,axiom,
    ( one_one_real
    = ( real2
      @ ^ [N3: nat] : one_one_rat ) ) ).

% one_real_def
thf(fact_9798_of__int__Real,axiom,
    ( ring_1_of_int_real
    = ( ^ [X3: int] :
          ( real2
          @ ^ [N3: nat] : ( ring_1_of_int_rat @ X3 ) ) ) ) ).

% of_int_Real
thf(fact_9799_cauchy__inverse,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ( cauchy
          @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X6 @ N3 ) ) ) ) ) ).

% cauchy_inverse
thf(fact_9800_minus__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( uminus_uminus_real @ ( real2 @ X6 ) )
        = ( real2
          @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X6 @ N3 ) ) ) ) ) ).

% minus_Real
thf(fact_9801_add__Real,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( ( plus_plus_real @ ( real2 @ X6 ) @ ( real2 @ Y6 ) )
          = ( real2
            @ ^ [N3: nat] : ( plus_plus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ) ).

% add_Real
thf(fact_9802_diff__Real,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( ( minus_minus_real @ ( real2 @ X6 ) @ ( real2 @ Y6 ) )
          = ( real2
            @ ^ [N3: nat] : ( minus_minus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ) ).

% diff_Real
thf(fact_9803_vanishes__mult__bounded,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ? [A8: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ A8 )
          & ! [N: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N ) ) @ A8 ) )
     => ( ( vanishes @ Y6 )
       => ( vanishes
          @ ^ [N3: nat] : ( times_times_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) ) ) ) ).

% vanishes_mult_bounded
thf(fact_9804_vanishes__diff__inverse,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ( ( cauchy @ Y6 )
         => ( ~ ( vanishes @ Y6 )
           => ( ( vanishes
                @ ^ [N3: nat] : ( minus_minus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) )
             => ( vanishes
                @ ^ [N3: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X6 @ N3 ) ) @ ( inverse_inverse_rat @ ( Y6 @ N3 ) ) ) ) ) ) ) ) ) ).

% vanishes_diff_inverse
thf(fact_9805_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X7: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K3 @ N3 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N3 ) ) @ R5 ) ) ) ) ) ).

% vanishes_def
thf(fact_9806_vanishesI,axiom,
    ! [X6: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [N: nat] :
              ( ( ord_less_eq_nat @ K4 @ N )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N ) ) @ R3 ) ) )
     => ( vanishes @ X6 ) ) ).

% vanishesI
thf(fact_9807_vanishesD,axiom,
    ! [X6: nat > rat,R2: rat] :
      ( ( vanishes @ X6 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [N7: nat] :
            ( ( ord_less_eq_nat @ K2 @ N7 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N7 ) ) @ R2 ) ) ) ) ).

% vanishesD
thf(fact_9808_cauchy__not__vanishes__cases,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ? [B4: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B4 )
            & ? [K2: nat] :
                ( ! [N7: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N7 )
                   => ( ord_less_rat @ B4 @ ( uminus_uminus_rat @ ( X6 @ N7 ) ) ) )
                | ! [N7: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N7 )
                   => ( ord_less_rat @ B4 @ ( X6 @ N7 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
thf(fact_9809_not__positive__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( ~ ( positive2 @ ( real2 @ X6 ) ) )
        = ( ! [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_eq_rat @ ( X6 @ N3 ) @ R5 ) ) ) ) ) ) ).

% not_positive_Real
thf(fact_9810_positive__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( positive2 @ ( real2 @ X6 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_rat @ R5 @ ( X6 @ N3 ) ) ) ) ) ) ) ).

% positive_Real
thf(fact_9811_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( ( positive2 @ X )
     => ( ( positive2 @ Y )
       => ( positive2 @ ( times_times_real @ X @ Y ) ) ) ) ).

% Real.positive_mult
thf(fact_9812_Real_Opositive__zero,axiom,
    ~ ( positive2 @ zero_zero_real ) ).

% Real.positive_zero
thf(fact_9813_Real_Opositive__add,axiom,
    ! [X: real,Y: real] :
      ( ( positive2 @ X )
     => ( ( positive2 @ Y )
       => ( positive2 @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% Real.positive_add
thf(fact_9814_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ ( positive2 @ X )
     => ( ( X != zero_zero_real )
       => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).

% Real.positive_minus
thf(fact_9815_less__real__def,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] : ( positive2 @ ( minus_minus_real @ Y2 @ X3 ) ) ) ) ).

% less_real_def
thf(fact_9816_Real_Opositive_Orep__eq,axiom,
    ( positive2
    = ( ^ [X3: real] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K3 @ N3 )
             => ( ord_less_rat @ R5 @ ( rep_real @ X3 @ N3 ) ) ) ) ) ) ).

% Real.positive.rep_eq
thf(fact_9817_inverse__real_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( inverse_inverse_real @ ( real2 @ X ) )
        = ( real2
          @ ( if_nat_rat @ ( vanishes @ X )
            @ ^ [N3: nat] : zero_zero_rat
            @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X @ N3 ) ) ) ) ) ) ).

% inverse_real.abs_eq
thf(fact_9818_realrel__refl,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( realrel @ X6 @ X6 ) ) ).

% realrel_refl
thf(fact_9819_one__real_Orsp,axiom,
    ( realrel
    @ ^ [N3: nat] : one_one_rat
    @ ^ [N3: nat] : one_one_rat ) ).

% one_real.rsp
thf(fact_9820_zero__real_Orsp,axiom,
    ( realrel
    @ ^ [N3: nat] : zero_zero_rat
    @ ^ [N3: nat] : zero_zero_rat ) ).

% zero_real.rsp
thf(fact_9821_real_Oabs__induct,axiom,
    ! [P: real > $o,X: real] :
      ( ! [Y3: nat > rat] :
          ( ( realrel @ Y3 @ Y3 )
         => ( P @ ( real2 @ Y3 ) ) )
     => ( P @ X ) ) ).

% real.abs_induct
thf(fact_9822_uminus__real_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( uminus_uminus_real @ ( real2 @ X ) )
        = ( real2
          @ ^ [N3: nat] : ( uminus_uminus_rat @ ( X @ N3 ) ) ) ) ) ).

% uminus_real.abs_eq
thf(fact_9823_plus__real_Oabs__eq,axiom,
    ! [Xa2: nat > rat,X: nat > rat] :
      ( ( realrel @ Xa2 @ Xa2 )
     => ( ( realrel @ X @ X )
       => ( ( plus_plus_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
          = ( real2
            @ ^ [N3: nat] : ( plus_plus_rat @ ( Xa2 @ N3 ) @ ( X @ N3 ) ) ) ) ) ) ).

% plus_real.abs_eq
thf(fact_9824_times__real_Oabs__eq,axiom,
    ! [Xa2: nat > rat,X: nat > rat] :
      ( ( realrel @ Xa2 @ Xa2 )
     => ( ( realrel @ X @ X )
       => ( ( times_times_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
          = ( real2
            @ ^ [N3: nat] : ( times_times_rat @ ( Xa2 @ N3 ) @ ( X @ N3 ) ) ) ) ) ) ).

% times_real.abs_eq
thf(fact_9825_realrelI,axiom,
    ! [X6: nat > rat,Y6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y6 )
       => ( ( vanishes
            @ ^ [N3: nat] : ( minus_minus_rat @ ( X6 @ N3 ) @ ( Y6 @ N3 ) ) )
         => ( realrel @ X6 @ Y6 ) ) ) ) ).

% realrelI
thf(fact_9826_realrel__def,axiom,
    ( realrel
    = ( ^ [X7: nat > rat,Y7: nat > rat] :
          ( ( cauchy @ X7 )
          & ( cauchy @ Y7 )
          & ( vanishes
            @ ^ [N3: nat] : ( minus_minus_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).

% realrel_def
thf(fact_9827_Real_Opositive_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( positive2 @ ( real2 @ X ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_rat @ R5 @ ( X @ N3 ) ) ) ) ) ) ) ).

% Real.positive.abs_eq
thf(fact_9828_Real_Opositive__def,axiom,
    ( positive2
    = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
      @ ^ [X7: nat > rat] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K3 @ N3 )
             => ( ord_less_rat @ R5 @ ( X7 @ N3 ) ) ) ) ) ) ).

% Real.positive_def
thf(fact_9829_inverse__real__def,axiom,
    ( inverse_inverse_real
    = ( map_fu7146612038024189824t_real @ rep_real @ real2
      @ ^ [X7: nat > rat] :
          ( if_nat_rat @ ( vanishes @ X7 )
          @ ^ [N3: nat] : zero_zero_rat
          @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X7 @ N3 ) ) ) ) ) ).

% inverse_real_def
thf(fact_9830_uminus__real__def,axiom,
    ( uminus_uminus_real
    = ( map_fu7146612038024189824t_real @ rep_real @ real2
      @ ^ [X7: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X7 @ N3 ) ) ) ) ).

% uminus_real_def
thf(fact_9831_times__real__def,axiom,
    ( times_times_real
    = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
      @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ).

% times_real_def
thf(fact_9832_plus__real__def,axiom,
    ( plus_plus_real
    = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
      @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ).

% plus_real_def
thf(fact_9833_Real_Opositive_Orsp,axiom,
    ( bNF_re728719798268516973at_o_o @ realrel
    @ ^ [Y5: $o,Z3: $o] : Y5 = Z3
    @ ^ [X7: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
           => ( ord_less_rat @ R5 @ ( X7 @ N3 ) ) ) )
    @ ^ [X7: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
           => ( ord_less_rat @ R5 @ ( X7 @ N3 ) ) ) ) ) ).

% Real.positive.rsp
thf(fact_9834_cr__real__def,axiom,
    ( cr_real
    = ( ^ [X3: nat > rat,Y2: real] :
          ( ( realrel @ X3 @ X3 )
          & ( ( real2 @ X3 )
            = Y2 ) ) ) ) ).

% cr_real_def
thf(fact_9835_plus__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) )
    @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) ) ) ).

% plus_real.rsp
thf(fact_9836_uminus__real_Orsp,axiom,
    ( bNF_re895249473297799549at_rat @ realrel @ realrel
    @ ^ [X7: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X7 @ N3 ) )
    @ ^ [X7: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X7 @ N3 ) ) ) ).

% uminus_real.rsp
thf(fact_9837_times__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) )
    @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) ) ) ).

% times_real.rsp
thf(fact_9838_gcd__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ ^ [Y5: int,Z3: int] : Y5 = Z3 )
    @ gcd_gcd_int
    @ gcd_gcd_int ) ).

% gcd_integer.rsp
thf(fact_9839_inverse__real_Orsp,axiom,
    ( bNF_re895249473297799549at_rat @ realrel @ realrel
    @ ^ [X7: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X7 )
        @ ^ [N3: nat] : zero_zero_rat
        @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X7 @ N3 ) ) )
    @ ^ [X7: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X7 )
        @ ^ [N3: nat] : zero_zero_rat
        @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X7 @ N3 ) ) ) ) ).

% inverse_real.rsp
thf(fact_9840_Real_Opositive_Otransfer,axiom,
    ( bNF_re4297313714947099218al_o_o @ pcr_real
    @ ^ [Y5: $o,Z3: $o] : Y5 = Z3
    @ ^ [X7: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
           => ( ord_less_rat @ R5 @ ( X7 @ N3 ) ) ) )
    @ positive2 ) ).

% Real.positive.transfer
thf(fact_9841_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X3: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X3 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_9842_real_Orel__eq__transfer,axiom,
    ( bNF_re4521903465945308077real_o @ pcr_real
    @ ( bNF_re4297313714947099218al_o_o @ pcr_real
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ realrel
    @ ^ [Y5: real,Z3: real] : Y5 = Z3 ) ).

% real.rel_eq_transfer
thf(fact_9843_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
thf(fact_9844_zero__real_Otransfer,axiom,
    ( pcr_real
    @ ^ [N3: nat] : zero_zero_rat
    @ zero_zero_real ) ).

% zero_real.transfer
thf(fact_9845_one__real_Otransfer,axiom,
    ( pcr_real
    @ ^ [N3: nat] : one_one_rat
    @ one_one_real ) ).

% one_real.transfer
thf(fact_9846_cr__real__eq,axiom,
    ( pcr_real
    = ( ^ [X3: nat > rat,Y2: real] :
          ( ( cauchy @ X3 )
          & ( ( real2 @ X3 )
            = Y2 ) ) ) ) ).

% cr_real_eq
thf(fact_9847_uminus__real_Otransfer,axiom,
    ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
    @ ^ [X7: nat > rat,N3: nat] : ( uminus_uminus_rat @ ( X7 @ N3 ) )
    @ uminus_uminus_real ) ).

% uminus_real.transfer
thf(fact_9848_plus__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( plus_plus_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) )
    @ plus_plus_real ) ).

% plus_real.transfer
thf(fact_9849_times__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X7: nat > rat,Y7: nat > rat,N3: nat] : ( times_times_rat @ ( X7 @ N3 ) @ ( Y7 @ N3 ) )
    @ times_times_real ) ).

% times_real.transfer
thf(fact_9850_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_9851_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ pcr_rat )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_9852_inverse__real_Otransfer,axiom,
    ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
    @ ^ [X7: nat > rat] :
        ( if_nat_rat @ ( vanishes @ X7 )
        @ ^ [N3: nat] : zero_zero_rat
        @ ^ [N3: nat] : ( inverse_inverse_rat @ ( X7 @ N3 ) ) )
    @ inverse_inverse_real ) ).

% inverse_real.transfer
thf(fact_9853_Rat_Opositive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y5: $o,Z3: $o] : Y5 = Z3
    @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
    @ positive ) ).

% Rat.positive.transfer
thf(fact_9854_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_9855_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_9856_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_9857_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ pcr_int
    @ ^ [N3: nat] : ( product_Pair_nat_nat @ N3 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9858_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_9859_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_9860_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_9861_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_9862_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_9863_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_9864_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_9865_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X @ V )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_9866_int_Orel__eq__transfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ intrel
    @ ^ [Y5: int,Z3: int] : Y5 = Z3 ) ).

% int.rel_eq_transfer
thf(fact_9867_int_Obi__total,axiom,
    bi_tot896582865486249351at_int @ pcr_int ).

% int.bi_total
thf(fact_9868_zero__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).

% zero_int.rsp
thf(fact_9869_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( abs_Integ @ X )
        = ( abs_Integ @ Y ) )
      = ( intrel @ X @ Y ) ) ).

% int.abs_eq_iff
thf(fact_9870_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X3: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X3 ) ) ) ).

% uminus_int.rsp
thf(fact_9871_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_9872_one__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).

% one_int.rsp
thf(fact_9873_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] :
              ( ( plus_plus_nat @ X3 @ V4 )
              = ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% intrel_def
thf(fact_9874_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% less_int.rsp
thf(fact_9875_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_9876_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_9877_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_9878_le__enumerate,axiom,
    ! [S3: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ).

% le_enumerate
thf(fact_9879_finite__le__enumerate,axiom,
    ! [S3: set_nat,N2: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
       => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ) ).

% finite_le_enumerate
thf(fact_9880_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% less_eq
thf(fact_9881_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_9882_Least__Suc2,axiom,
    ! [P: nat > $o,N2: nat,Q: nat > $o,M: nat] :
      ( ( P @ N2 )
     => ( ( Q @ M )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P @ ( suc @ K2 ) )
                = ( Q @ K2 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_9883_Least__Suc,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M6: nat] : ( P @ ( suc @ M6 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_9884_Sup__real__def,axiom,
    ( comple1385675409528146559p_real
    = ( ^ [X7: set_real] :
          ( ord_Least_real
          @ ^ [Z5: real] :
            ! [X3: real] :
              ( ( member_real @ X3 @ X7 )
             => ( ord_less_eq_real @ X3 @ Z5 ) ) ) ) ) ).

% Sup_real_def
thf(fact_9885_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_9886_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_9887_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X3: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X3 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
    @ ^ [X3: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X3 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_9888_csqrt__0,axiom,
    ( ( csqrt @ zero_zero_complex )
    = zero_zero_complex ) ).

% csqrt_0
thf(fact_9889_csqrt__eq__0,axiom,
    ! [Z2: complex] :
      ( ( ( csqrt @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_complex ) ) ).

% csqrt_eq_0
thf(fact_9890_ratrel__iff,axiom,
    ( ratrel
    = ( ^ [X3: product_prod_int_int,Y2: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X3 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y2 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).

% ratrel_iff
thf(fact_9891_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_9892_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ ratrel )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).

% Fract.rsp
thf(fact_9893_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X3: product_prod_int_int,Y2: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X3 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y2 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y2 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).

% ratrel_def
thf(fact_9894_Rat_Opositive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y5: $o,Z3: $o] : Y5 = Z3
    @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
    @ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).

% Rat.positive.rsp
thf(fact_9895_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_9896_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( positive @ ( abs_Rat @ X ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive.abs_eq
thf(fact_9897_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_9898_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_9899_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X3: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X3 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_9900_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z5: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z5 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9901_complex__Im__fact,axiom,
    ! [N2: nat] :
      ( ( im @ ( semiri5044797733671781792omplex @ N2 ) )
      = zero_zero_real ) ).

% complex_Im_fact
thf(fact_9902_complex__Im__of__int,axiom,
    ! [Z2: int] :
      ( ( im @ ( ring_17405671764205052669omplex @ Z2 ) )
      = zero_zero_real ) ).

% complex_Im_of_int
thf(fact_9903_complex__Im__of__nat,axiom,
    ! [N2: nat] :
      ( ( im @ ( semiri8010041392384452111omplex @ N2 ) )
      = zero_zero_real ) ).

% complex_Im_of_nat
thf(fact_9904_Im__complex__of__real,axiom,
    ! [Z2: real] :
      ( ( im @ ( real_V4546457046886955230omplex @ Z2 ) )
      = zero_zero_real ) ).

% Im_complex_of_real
thf(fact_9905_Im__power__real,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X @ N2 ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_9906_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_9907_Re__power__real,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N2 ) )
        = ( power_power_real @ ( re @ X ) @ N2 ) ) ) ).

% Re_power_real
thf(fact_9908_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( ( re @ Y )
          = zero_zero_real )
       => ( ( cos_real @ ( arg @ Y ) )
          = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
thf(fact_9909_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
       => ( ( csqrt @ X )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9910_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_9911_zero__complex_Osimps_I1_J,axiom,
    ( ( re @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(1)
thf(fact_9912_cmod__eq__Re,axiom,
    ! [Z2: complex] :
      ( ( ( im @ Z2 )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z2 )
        = ( abs_abs_real @ ( re @ Z2 ) ) ) ) ).

% cmod_eq_Re
thf(fact_9913_cmod__eq__Im,axiom,
    ! [Z2: complex] :
      ( ( ( re @ Z2 )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z2 )
        = ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).

% cmod_eq_Im
thf(fact_9914_Im__eq__0,axiom,
    ! [Z2: complex] :
      ( ( ( abs_abs_real @ ( re @ Z2 ) )
        = ( real_V1022390504157884413omplex @ Z2 ) )
     => ( ( im @ Z2 )
        = zero_zero_real ) ) ).

% Im_eq_0
thf(fact_9915_complex__is__Int__iff,axiom,
    ! [Z2: complex] :
      ( ( member_complex @ Z2 @ ring_1_Ints_complex )
      = ( ( ( im @ Z2 )
          = zero_zero_real )
        & ? [I3: int] :
            ( ( re @ Z2 )
            = ( ring_1_of_int_real @ I3 ) ) ) ) ).

% complex_is_Int_iff
thf(fact_9916_zero__complex_Osimps_I2_J,axiom,
    ( ( im @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(2)
thf(fact_9917_complex__is__Nat__iff,axiom,
    ! [Z2: complex] :
      ( ( member_complex @ Z2 @ semiri3842193898606819883omplex )
      = ( ( ( im @ Z2 )
          = zero_zero_real )
        & ? [I3: nat] :
            ( ( re @ Z2 )
            = ( semiri5074537144036343181t_real @ I3 ) ) ) ) ).

% complex_is_Nat_iff
thf(fact_9918_csqrt__principal,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z2 ) ) )
      | ( ( ( re @ ( csqrt @ Z2 ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z2 ) ) ) ) ) ).

% csqrt_principal
thf(fact_9919_Re__csqrt,axiom,
    ! [Z2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z2 ) ) ) ).

% Re_csqrt
thf(fact_9920_complex__eq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_9921_complex__neq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_9922_csqrt__square,axiom,
    ! [B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
        | ( ( ( re @ B )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B ) ) ).

% csqrt_square
thf(fact_9923_csqrt__unique,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z2 )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
          | ( ( ( re @ W2 )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
       => ( ( csqrt @ Z2 )
          = W2 ) ) ) ).

% csqrt_unique
thf(fact_9924_complex__unit__circle,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_9925_cmod__plus__Re__le__0__iff,axiom,
    ! [Z2: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
      = ( ( re @ Z2 )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_9926_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] :
      ( ( im @ ( csqrt @ Z2 ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z2 )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_9927_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9928_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_9929_imaginary__unit_Osimps_I1_J,axiom,
    ( ( re @ imaginary_unit )
    = zero_zero_real ) ).

% imaginary_unit.simps(1)
thf(fact_9930_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
thf(fact_9931_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( ( complex2 @ X @ Y )
        = imaginary_unit )
      = ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9932_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9933_complex__of__real__i,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% complex_of_real_i
thf(fact_9934_i__complex__of__real,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% i_complex_of_real
thf(fact_9935_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9936_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9937_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9938_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9939_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9940_Complex__in__Reals,axiom,
    ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).

% Complex_in_Reals
thf(fact_9941_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9942_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_9943_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_9944_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9945_gcd__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( gcd_gcd_int @ Xa2 @ X ) ) ) ).

% gcd_integer.abs_eq
thf(fact_9946_gcd__code__integer,axiom,
    ( gcd_gcd_Code_integer
    = ( ^ [K3: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).

% gcd_code_integer
thf(fact_9947_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_9948_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9949_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_9950_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa2 @ X ) ) ).

% less_integer.abs_eq
thf(fact_9951_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9952_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_9953_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9954_complex__is__Real__iff,axiom,
    ! [Z2: complex] :
      ( ( member_complex @ Z2 @ real_V2521375963428798218omplex )
      = ( ( im @ Z2 )
        = zero_zero_real ) ) ).

% complex_is_Real_iff
thf(fact_9955_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_9956_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9957_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_9958_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9959_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_9960_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
      @ ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_9961_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y5: int,Z3: int] : Y5 = Z3
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y5: int,Z3: int] : Y5 = Z3
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_9962_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9963_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9964_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9965_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_9966_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_iff
thf(fact_9967_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9968_gcd__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X @ Xa2 ) )
      = ( gcd_gcd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% gcd_integer.rep_eq
thf(fact_9969_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9970_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9971_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_9972_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_9973_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_9974_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_9975_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_9976_nat__of__integer__code__post_I2_J,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_code_post(2)
thf(fact_9977_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S5 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9978_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S5 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S5 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9979_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_9980_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X3: real,Y2: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y2 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_9981_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X3: complex,Y2: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y2 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_9982_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: nat > real,M7: nat] :
      ( ( bfun_nat_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M7 ) )
        @ at_top_nat )
     => ( ! [M5: nat,N: nat] :
            ( ( ord_less_eq_nat @ M7 @ M5 )
           => ( ( ord_less_eq_nat @ M5 @ N )
             => ( ord_less_eq_real @ ( F @ M5 ) @ ( F @ N ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_inc
thf(fact_9983_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: nat > real,M7: nat] :
      ( ( bfun_nat_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M7 ) )
        @ at_top_nat )
     => ( ! [M5: nat,N: nat] :
            ( ( ord_less_eq_nat @ M7 @ M5 )
           => ( ( ord_less_eq_nat @ M5 @ N )
             => ( ord_less_eq_real @ ( F @ N ) @ ( F @ M5 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_dec
thf(fact_9984_Bseq__mono__convergent,axiom,
    ! [X6: nat > real] :
      ( ( bfun_nat_real @ X6 @ at_top_nat )
     => ( ! [M5: nat,N: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
           => ( ord_less_eq_real @ ( X6 @ M5 ) @ ( X6 @ N ) ) )
       => ( topolo7531315842566124627t_real @ X6 ) ) ) ).

% Bseq_mono_convergent
thf(fact_9985_convergent__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).

% convergent_realpow
thf(fact_9986_at__right__to__0,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
      = ( filtermap_real_real
        @ ^ [X3: real] : ( plus_plus_real @ X3 @ A )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_right_to_0
thf(fact_9987_at__top__to__right,axiom,
    ( at_top_real
    = ( filtermap_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_top_to_right
thf(fact_9988_at__right__to__top,axiom,
    ( ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
    = ( filtermap_real_real @ inverse_inverse_real @ at_top_real ) ) ).

% at_right_to_top
thf(fact_9989_filtermap__ln__at__right,axiom,
    ( ( filtermap_real_real @ ln_ln_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
    = at_bot_real ) ).

% filtermap_ln_at_right
thf(fact_9990_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M6: nat,N3: nat] :
        ( ( dvd_dvd_nat @ M6 @ N3 )
        & ( M6 != N3 ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_9991_binomial__def,axiom,
    ( binomial
    = ( ^ [N3: 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 @ N3 ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9992_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X3 )
    @ ^ [X3: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X3 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_9993_set__encode__vimage__Suc,axiom,
    ! [A3: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A3 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc
thf(fact_9994_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
      = ( vimage_nat_nat @ suc @ A3 ) ) ).

% vimage_Suc_insert_0
thf(fact_9995_vimage__Suc__insert__Suc,axiom,
    ! [N2: nat,A3: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N2 ) @ A3 ) )
      = ( insert_nat @ N2 @ ( vimage_nat_nat @ suc @ A3 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_9996_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_9997_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_9998_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ( euclid3395696857347342551nt_int @ K )
        = ( sgn_sgn_int @ K ) ) ) ).

% division_segment_eq_sgn
thf(fact_9999_division__segment__nat__def,axiom,
    ( euclid3398187327856392827nt_nat
    = ( ^ [N3: nat] : one_one_nat ) ) ).

% division_segment_nat_def
thf(fact_10000_division__segment__int__def,axiom,
    ( euclid3395696857347342551nt_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% division_segment_int_def
thf(fact_10001_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M6: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M6 )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_10002_less__enat__def,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [M6: extended_enat,N3: extended_enat] :
          ( extended_case_enat_o
          @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N3 )
          @ $false
          @ M6 ) ) ) ).

% less_enat_def
thf(fact_10003_transp__realrel,axiom,
    transp_nat_rat @ realrel ).

% transp_realrel
thf(fact_10004_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_10005_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_10006_of__rat__Real,axiom,
    ( field_7254667332652039916t_real
    = ( ^ [X3: rat] :
          ( real2
          @ ^ [N3: nat] : X3 ) ) ) ).

% of_rat_Real
thf(fact_10007_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Q2: rat] :
          ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q2 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q2 ) @ Y ) ) ) ).

% of_rat_dense
thf(fact_10008_less__RealD,axiom,
    ! [Y6: nat > rat,X: real] :
      ( ( cauchy @ Y6 )
     => ( ( ord_less_real @ X @ ( real2 @ Y6 ) )
       => ? [N: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y6 @ N ) ) ) ) ) ).

% less_RealD
thf(fact_10009_Real__leI,axiom,
    ! [X6: nat > rat,Y: real] :
      ( ( cauchy @ X6 )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X6 @ N ) ) @ Y )
       => ( ord_less_eq_real @ ( real2 @ X6 ) @ Y ) ) ) ).

% Real_leI
thf(fact_10010_le__RealI,axiom,
    ! [Y6: nat > rat,X: real] :
      ( ( cauchy @ Y6 )
     => ( ! [N: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y6 @ N ) ) )
       => ( ord_less_eq_real @ X @ ( real2 @ Y6 ) ) ) ) ).

% le_RealI
thf(fact_10011_compute__powr__real,axiom,
    ( powr_real2
    = ( ^ [B2: real,I3: real] :
          ( if_real @ ( ord_less_eq_real @ B2 @ zero_zero_real )
          @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ ^ [Uu3: product_unit] : ( powr_real2 @ B2 @ I3 ) )
          @ ( if_real
            @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I3 ) )
              = I3 )
            @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I3 ) @ ( power_power_real @ B2 @ ( nat2 @ ( archim6058952711729229775r_real @ I3 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B2 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I3 ) ) ) ) ) )
            @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ^ [Uu3: product_unit] : ( powr_real2 @ B2 @ I3 ) ) ) ) ) ) ).

% compute_powr_real
thf(fact_10012_String_Oempty__neq__Literal,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,S: literal] :
      ( zero_zero_literal
     != ( literal2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ S ) ) ).

% String.empty_neq_Literal
thf(fact_10013_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_10014_UNIV__char__of__nat,axiom,
    ( top_top_set_char
    = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_10015_range__nat__of__char,axiom,
    ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
    = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_10016_char_Osize_I2_J,axiom,
    ! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X15 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_10017_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_10018_char_Osize__gen,axiom,
    ! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X15 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_10019_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N5: nat] :
          ! [M6: nat] :
            ( ( ord_less_eq_nat @ N5 @ M6 )
           => ! [N3: nat] :
                ( ( ord_less_eq_nat @ N5 @ N3 )
               => ( P @ ( product_Pair_nat_nat @ N3 @ M6 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10020_numeral__le__enat__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% numeral_le_enat_iff
thf(fact_10021_elimnum,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimnum
thf(fact_10022_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% enat_ord_simps(2)
thf(fact_10023_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% enat_ord_simps(1)
thf(fact_10024_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% plus_enat_simps(1)
thf(fact_10025_idiff__enat__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N2 )
      = ( extended_enat2 @ zero_zero_nat ) ) ).

% idiff_enat_0
thf(fact_10026_idiff__enat__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ ( extended_enat2 @ zero_zero_nat ) )
      = N2 ) ).

% idiff_enat_0_right
thf(fact_10027_numeral__less__enat__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% numeral_less_enat_iff
thf(fact_10028_enat__iless,axiom,
    ! [N2: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N2 @ ( extended_enat2 @ M ) )
     => ? [K2: nat] :
          ( N2
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_iless
thf(fact_10029_less__enatE,axiom,
    ! [N2: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N2 @ ( extended_enat2 @ M ) )
     => ~ ! [K2: nat] :
            ( ( N2
              = ( extended_enat2 @ K2 ) )
           => ~ ( ord_less_nat @ K2 @ M ) ) ) ).

% less_enatE
thf(fact_10030_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Uu: extended_enat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A @ B ) @ Uu )
      = ( vEBT_Leaf @ A @ B ) ) ).

% VEBT_internal.elim_dead.simps(1)
thf(fact_10031_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_on7984719198319812577d_enat
        = ( extended_enat2 @ X ) )
      = ( X = one_one_nat ) ) ).

% enat_1_iff(2)
thf(fact_10032_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = one_on7984719198319812577d_enat )
      = ( X = one_one_nat ) ) ).

% enat_1_iff(1)
thf(fact_10033_one__enat__def,axiom,
    ( one_on7984719198319812577d_enat
    = ( extended_enat2 @ one_one_nat ) ) ).

% one_enat_def
thf(fact_10034_Suc__ile__eq,axiom,
    ! [M: nat,N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N2 )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N2 ) ) ).

% Suc_ile_eq
thf(fact_10035_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X ) )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_10036_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = zero_z5237406670263579293d_enat )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_10037_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_10038_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L ) )
      = ( vEBT_Node @ Info2 @ Deg
        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( map_VE8901447254227204932T_VEBT
            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
            @ TreeList2 ) )
        @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.simps(3)
thf(fact_10039_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N2 ) )
      = ( ? [Y8: nat,X9: nat] :
            ( ( X
              = ( extended_enat2 @ X9 ) )
            & ( Y
              = ( extended_enat2 @ Y8 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N2 ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_10040_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y
             != ( vEBT_Leaf @ A4 @ B4 ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y
                 != ( vEBT_Node @ Info @ Deg2
                    @ ( map_VE8901447254227204932T_VEBT
                      @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      @ TreeList )
                    @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
               => ! [L4: nat] :
                    ( ( Xa2
                      = ( extended_enat2 @ L4 ) )
                   => ( Y
                     != ( vEBT_Node @ Info @ Deg2
                        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ( map_VE8901447254227204932T_VEBT
                            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            @ TreeList ) )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
thf(fact_10041_elimcomplete,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimcomplete
thf(fact_10042_enat__ord__simps_I4_J,axiom,
    ! [Q4: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Q4 @ extend5688581933313929465d_enat )
      = ( Q4 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
thf(fact_10043_enat__ord__simps_I6_J,axiom,
    ! [Q4: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q4 ) ).

% enat_ord_simps(6)
thf(fact_10044_idiff__self,axiom,
    ! [N2: extended_enat] :
      ( ( N2 != extend5688581933313929465d_enat )
     => ( ( minus_3235023915231533773d_enat @ N2 @ N2 )
        = zero_z5237406670263579293d_enat ) ) ).

% idiff_self
thf(fact_10045_idiff__infinity__right,axiom,
    ! [A: nat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ A ) @ extend5688581933313929465d_enat )
      = zero_z5237406670263579293d_enat ) ).

% idiff_infinity_right
thf(fact_10046_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
          = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
thf(fact_10047_times__enat__simps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N2 ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N2 ) )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
thf(fact_10048_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A4: $o,B4: $o,Uu2: extended_enat] :
          ( X
         != ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A4 @ B4 ) @ Uu2 ) )
     => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ extend5688581933313929465d_enat ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,L4: nat] :
              ( X
             != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).

% VEBT_internal.elim_dead.cases
thf(fact_10049_enat__ord__code_I4_J,axiom,
    ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).

% enat_ord_code(4)
thf(fact_10050_less__infinityE,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ N2 @ extend5688581933313929465d_enat )
     => ~ ! [K2: nat] :
            ( N2
           != ( extended_enat2 @ K2 ) ) ) ).

% less_infinityE
thf(fact_10051_infinity__ilessE,axiom,
    ! [M: nat] :
      ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).

% infinity_ilessE
thf(fact_10052_Sup__enat__def,axiom,
    ( comple4398354569131411667d_enat
    = ( ^ [A7: set_Extended_enat] : ( if_Extended_enat @ ( A7 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A7 ) @ ( lattic921264341876707157d_enat @ A7 ) @ extend5688581933313929465d_enat ) ) ) ) ).

% Sup_enat_def
thf(fact_10053_infinity__ne__i0,axiom,
    extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).

% infinity_ne_i0
thf(fact_10054_imult__is__infinity,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B )
        = extend5688581933313929465d_enat )
      = ( ( ( A = extend5688581933313929465d_enat )
          & ( B != zero_z5237406670263579293d_enat ) )
        | ( ( B = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
thf(fact_10055_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( ( A != extend5688581933313929465d_enat )
        & ( ord_le72135733267957522d_enat @ B @ C ) ) ) ).

% enat_add_left_cancel_less
thf(fact_10056_imult__infinity,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
     => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N2 )
        = extend5688581933313929465d_enat ) ) ).

% imult_infinity
thf(fact_10057_imult__infinity__right,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
     => ( ( times_7803423173614009249d_enat @ N2 @ extend5688581933313929465d_enat )
        = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
thf(fact_10058_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
      = ( vEBT_Node @ Info2 @ Deg
        @ ( map_VE8901447254227204932T_VEBT
          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          @ TreeList2 )
        @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).

% VEBT_internal.elim_dead.simps(2)
thf(fact_10059_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M6: extended_enat,N3: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P5: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P5 ) )
              @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
              @ N3 )
          @ ( if_Extended_enat @ ( N3 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M6 ) ) ) ).

% times_enat_def
thf(fact_10060_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y
                  = ( vEBT_Leaf @ A4 @ B4 ) )
               => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y
                      = ( vEBT_Node @ Info @ Deg2
                        @ ( map_VE8901447254227204932T_VEBT
                          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ TreeList )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
                   => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
           => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
                 => ! [L4: nat] :
                      ( ( Xa2
                        = ( extended_enat2 @ L4 ) )
                     => ( ( Y
                          = ( vEBT_Node @ Info @ Deg2
                            @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                              @ ( map_VE8901447254227204932T_VEBT
                                @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ TreeList ) )
                            @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
thf(fact_10061_plus__enat__def,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [M6: extended_enat,N3: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P5: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P5 ) )
              @ extend5688581933313929465d_enat
              @ N3 )
          @ extend5688581933313929465d_enat
          @ M6 ) ) ) ).

% plus_enat_def
thf(fact_10062_diff__enat__def,axiom,
    ( minus_3235023915231533773d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [X3: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [Y2: nat] : ( extended_enat2 @ ( minus_minus_nat @ X3 @ Y2 ) )
              @ zero_z5237406670263579293d_enat
              @ B2 )
          @ extend5688581933313929465d_enat
          @ A2 ) ) ) ).

% diff_enat_def
thf(fact_10063_eSuc__def,axiom,
    ( extended_eSuc
    = ( extend3600170679010898289d_enat
      @ ^ [N3: nat] : ( extended_enat2 @ ( suc @ N3 ) )
      @ extend5688581933313929465d_enat ) ) ).

% eSuc_def
thf(fact_10064_iless__Suc__eq,axiom,
    ! [M: nat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N2 ) )
      = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N2 ) ) ).

% iless_Suc_eq
thf(fact_10065_eSuc__mono,axiom,
    ! [N2: extended_enat,M: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N2 ) @ ( extended_eSuc @ M ) )
      = ( ord_le72135733267957522d_enat @ N2 @ M ) ) ).

% eSuc_mono
thf(fact_10066_iless__eSuc0,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ N2 @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
thf(fact_10067_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( ( extended_enat2 @ Y )
        = ( extended_eSuc @ X ) )
      = ( ? [N3: nat] :
            ( ( Y
              = ( suc @ N3 ) )
            & ( ( extended_enat2 @ N3 )
              = X ) ) ) ) ).

% enat_eSuc_iff
thf(fact_10068_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( ( extended_eSuc @ X )
        = ( extended_enat2 @ Y ) )
      = ( ? [N3: nat] :
            ( ( Y
              = ( suc @ N3 ) )
            & ( X
              = ( extended_enat2 @ N3 ) ) ) ) ) ).

% eSuc_enat_iff
thf(fact_10069_eSuc__enat,axiom,
    ! [N2: nat] :
      ( ( extended_eSuc @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( suc @ N2 ) ) ) ).

% eSuc_enat
thf(fact_10070_i0__iless__eSuc,axiom,
    ! [N2: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N2 ) ) ).

% i0_iless_eSuc
thf(fact_10071_one__eSuc,axiom,
    ( one_on7984719198319812577d_enat
    = ( extended_eSuc @ zero_z5237406670263579293d_enat ) ) ).

% one_eSuc
thf(fact_10072_zero__ne__eSuc,axiom,
    ! [N2: extended_enat] :
      ( zero_z5237406670263579293d_enat
     != ( extended_eSuc @ N2 ) ) ).

% zero_ne_eSuc
thf(fact_10073_not__eSuc__ilei0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_eSuc_ilei0
thf(fact_10074_ileI1,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M @ N2 )
     => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N2 ) ) ).

% ileI1
thf(fact_10075_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_10076_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_10077_Quotient__real,axiom,
    quotie3684837364556693515t_real @ realrel @ real2 @ rep_real @ cr_real ).

% Quotient_real
thf(fact_10078_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_10079_complex__cnj__zero,axiom,
    ( ( cnj @ zero_zero_complex )
    = zero_zero_complex ) ).

% complex_cnj_zero
thf(fact_10080_complex__cnj__zero__iff,axiom,
    ! [Z2: complex] :
      ( ( ( cnj @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
thf(fact_10081_complex__In__mult__cnj__zero,axiom,
    ! [Z2: complex] :
      ( ( im @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_10082_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_10083_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_10084_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_10085_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_10086_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_10087_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_10088_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_10089_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_10090_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_10091_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_10092_cr__int__def,axiom,
    ( cr_int
    = ( ^ [X3: product_prod_nat_nat] :
          ( ^ [Y5: int,Z3: int] : Y5 = Z3
          @ ( abs_Integ @ X3 ) ) ) ) ).

% cr_int_def
thf(fact_10093_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
thf(fact_10094_Quotient__int,axiom,
    quotie1194848508323700631at_int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).

% Quotient_int
thf(fact_10095_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.semilattice_neutr_axioms
thf(fact_10096_max__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.semilattice_neutr_axioms
thf(fact_10097_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_10098_Restr__natLeq,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N2 ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N2 ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_10099_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_10100_Restr__natLeq2,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
          @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_10101_natLeq__underS__less,axiom,
    ! [N2: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
      = ( collect_nat
        @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N2 ) ) ) ).

% natLeq_underS_less
thf(fact_10102_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_10103_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_10104_wf__int__ge__less__than2,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).

% wf_int_ge_less_than2
thf(fact_10105_wf__int__ge__less__than,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).

% wf_int_ge_less_than
thf(fact_10106_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_10107_max__nat_Omonoid__axioms,axiom,
    monoid_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.monoid_axioms
thf(fact_10108_gcd__nat_Omonoid__axioms,axiom,
    monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.monoid_axioms
thf(fact_10109_rcis__eq__zero__iff,axiom,
    ! [R2: real,A: real] :
      ( ( ( rcis @ R2 @ A )
        = zero_zero_complex )
      = ( R2 = zero_zero_real ) ) ).

% rcis_eq_zero_iff
thf(fact_10110_rcis__zero__mod,axiom,
    ! [A: real] :
      ( ( rcis @ zero_zero_real @ A )
      = zero_zero_complex ) ).

% rcis_zero_mod
thf(fact_10111_rcis__zero__arg,axiom,
    ! [R2: real] :
      ( ( rcis @ R2 @ zero_zero_real )
      = ( real_V4546457046886955230omplex @ R2 ) ) ).

% rcis_zero_arg
thf(fact_10112_Lcm__eq__0__I__nat,axiom,
    ! [A3: set_nat] :
      ( ( member_nat @ zero_zero_nat @ A3 )
     => ( ( gcd_Lcm_nat @ A3 )
        = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
thf(fact_10113_abs__Lcm__eq,axiom,
    ! [K5: set_int] :
      ( ( abs_abs_int @ ( gcd_Lcm_int @ K5 ) )
      = ( gcd_Lcm_int @ K5 ) ) ).

% abs_Lcm_eq
thf(fact_10114_Lcm__0__iff__nat,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( gcd_Lcm_nat @ A3 )
          = zero_zero_nat )
        = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).

% Lcm_0_iff_nat
thf(fact_10115_Lcm__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Lcm_int @ ( image_int_int @ abs_abs_int @ K5 ) )
      = ( gcd_Lcm_int @ K5 ) ) ).

% Lcm_abs_eq
thf(fact_10116_Lcm__int__eq,axiom,
    ! [N6: set_nat] :
      ( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N6 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N6 ) ) ) ).

% Lcm_int_eq
thf(fact_10117_Lcm__nat__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Lcm_nat
        @ ( image_int_nat
          @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
          @ K5 ) )
      = ( nat2 @ ( gcd_Lcm_int @ K5 ) ) ) ).

% Lcm_nat_abs_eq
thf(fact_10118_Gcd__nat__def,axiom,
    ( gcd_Gcd_nat
    = ( ^ [M9: set_nat] :
          ( gcd_Lcm_nat
          @ ( collect_nat
            @ ^ [D6: nat] :
              ! [X3: nat] :
                ( ( member_nat @ X3 @ M9 )
               => ( dvd_dvd_nat @ D6 @ X3 ) ) ) ) ) ) ).

% Gcd_nat_def
thf(fact_10119_dvd__Lcm__nat,axiom,
    ! [M: nat,M7: set_nat] :
      ( ( member_nat @ M @ M7 )
     => ( dvd_dvd_nat @ M @ ( gcd_Lcm_nat @ M7 ) ) ) ).

% dvd_Lcm_nat
thf(fact_10120_Lcm__dvd__nat,axiom,
    ! [M7: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ M7 )
         => ( dvd_dvd_nat @ X4 @ N2 ) )
     => ( dvd_dvd_nat @ ( gcd_Lcm_nat @ M7 ) @ N2 ) ) ).

% Lcm_dvd_nat
thf(fact_10121_Lcm__least__int,axiom,
    ! [A3: set_int,A: int] :
      ( ! [B4: int] :
          ( ( member_int @ B4 @ A3 )
         => ( dvd_dvd_int @ B4 @ A ) )
     => ( dvd_dvd_int @ ( gcd_Lcm_int @ A3 ) @ A ) ) ).

% Lcm_least_int
thf(fact_10122_dvd__Lcm__int,axiom,
    ! [M: int,M7: set_int] :
      ( ( member_int @ M @ M7 )
     => ( dvd_dvd_int @ M @ ( gcd_Lcm_int @ M7 ) ) ) ).

% dvd_Lcm_int
thf(fact_10123_Lcm__nat__infinite,axiom,
    ! [M7: set_nat] :
      ( ~ ( finite_finite_nat @ M7 )
     => ( ( gcd_Lcm_nat @ M7 )
        = zero_zero_nat ) ) ).

% Lcm_nat_infinite
thf(fact_10124_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_10125_Lcm__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K5 ) ) ).

% Lcm_int_greater_eq_0
thf(fact_10126_Lcm__int__def,axiom,
    ( gcd_Lcm_int
    = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).

% Lcm_int_def
thf(fact_10127_Lcm__int__set__eq__fold,axiom,
    ! [Xs2: list_int] :
      ( ( gcd_Lcm_int @ ( set_int2 @ Xs2 ) )
      = ( fold_int_int @ gcd_lcm_int @ Xs2 @ one_one_int ) ) ).

% Lcm_int_set_eq_fold
thf(fact_10128_Lcm__eq__Max__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ! [M5: nat,N: nat] :
                ( ( member_nat @ M5 @ M7 )
               => ( ( member_nat @ N @ M7 )
                 => ( member_nat @ ( gcd_lcm_nat @ M5 @ N ) @ M7 ) ) )
           => ( ( gcd_Lcm_nat @ M7 )
              = ( lattic8265883725875713057ax_nat @ M7 ) ) ) ) ) ) ).

% Lcm_eq_Max_nat

% Helper facts (42)
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,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X7: real] : ( P @ X7 ) ) ) ).

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__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $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__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $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__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: nat > rat,Y: nat > rat] :
      ( ( if_nat_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: nat > rat,Y: nat > rat] :
      ( ( if_nat_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $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_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 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).

%------------------------------------------------------------------------------