%------------------------------------------------------------------------------
% File     : ITP284_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_BuildupMemImp 00837_038274
% 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    : 0093_VEBT_BuildupMemImp_00837_038274 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13290 (4141 unt;2480 typ;   0 def)
%            Number of atoms       : 27158 (10904 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19214 (2866   ~; 527   |;1798   &)
%                                         (2765 <=>;11258  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of types       :  548 ( 547 usr)
%            Number of type conns  : 2450 (1522   >; 928   *;   0   +;   0  <<)
%            Number of predicates  :   84 (  81 usr;   2 prp; 0-3 aty)
%            Number of functors    : 1852 (1852 usr; 411 con; 0-8 aty)
%            Number of variables   : 27540 (26669   !; 871   ?;27540   :)
% SPC      : TF0_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 19:14:31.004
%------------------------------------------------------------------------------
% Could-be-implicit typings (547)
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tff(ty_n_t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    fun_nat_bool: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    fun_int_bool: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Nat__Onat_J,type,
    fun_bool_nat: $tType ).

tff(ty_n_t__fun_It__HOL__Obool_Mt__Int__Oint_J,type,
    fun_bool_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    fun_rat_rat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
    fun_num_num: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
    fun_num_nat: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    fun_num_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
    fun_nat_num: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    fun_int_rat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    fun_int_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    fun_int_int: $tType ).

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

tff(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
    option_real: $tType ).

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

tff(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
    set_literal: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
    itself_finite_2: $tType ).

tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

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

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

tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

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

tff(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_VEBTi: $tType ).

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

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

tff(ty_n_t__Heap__Oarray_It__HOL__Obool_J,type,
    array_bool: $tType ).

tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
    list_bool: $tType ).

tff(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
    array_nat: $tType ).

tff(ty_n_t__Heap__Oarray_It__Int__Oint_J,type,
    array_int: $tType ).

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

tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
    list_num: $tType ).

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

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

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

tff(ty_n_t__Set__Oset_It__HOL__Obool_J,type,
    set_bool: $tType ).

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

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

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

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

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

tff(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

tff(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

tff(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

tff(ty_n_t__Uint32__Ouint32,type,
    uint32: $tType ).

tff(ty_n_t__String__Ochar,type,
    char: $tType ).

tff(ty_n_t__Real__Oreal,type,
    real: $tType ).

tff(ty_n_t__HOL__Obool,type,
    bool: $tType ).

tff(ty_n_t__Rat__Orat,type,
    rat: $tType ).

tff(ty_n_t__Num__Onum,type,
    num: $tType ).

tff(ty_n_t__Nat__Onat,type,
    nat: $tType ).

tff(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (1933)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_complex > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_Pr958786334691620121nt_int > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_real_real ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_int_int ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_real_real ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_int_int ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: nat > fun_na1773021792790457535l_assn ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_VE1571388752190420990T_assn > fun_li2158447775985626393l_assn ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_VE4931415397359990460t_assn > fun_li7943157569623896045l_assn ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_VE123837653852366616t_assn > fun_li3347143843831059529l_assn ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_VE5577048379444288083T_assn > fun_li1808587207459806180l_assn ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_VE2561185598974572172i_assn > fun_li8369987531566349255l_assn ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_VE1225457145571040039t_assn > fun_li8822687668615115362l_assn ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_VE5641251438918192003t_assn > fun_li4226673942822278846l_assn ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_re8050108611700907676T_assn > fun_li7232704599769625645l_assn ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_re560840401387337987i_assn > fun_li2608363201533078078l_assn ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_re574865750389657758t_assn > fun_li204261821335639001l_assn ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_bool_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_bool_assn > fun_as6968651184343528707l_assn ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: assn > fun_bo8172996160782635326l_assn ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: assn > fun_op6483474040057813218t_assn ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: assn > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: assn > fun_VE2561185598974572172i_assn ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: nat > fun_li8672845860709154922t_assn ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: nat > fun_li8221786654549472865l_assn ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: nat > fun_li7646410839599160030t_assn ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: nat > fun_li6911922195568332558t_assn ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: nat > fun_li6689927339917241139i_assn ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_bool_assn > fun_as6968651184343528707l_assn ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_real > fun_na6137814669449589053l_bool ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_Pr5160692531401422620nt_nat > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_Pr5160692531401422620nt_nat > fun_fu7672510408924296255t_bool ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_nat_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_nat_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_int_nat > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_complex_nat > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_Pr5160692531401422620nt_nat > fun_Pr3099820731930151597t_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_int_nat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_complex_nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_Pr5160692531401422620nt_nat > fun_fu600552898087764240t_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_nat_nat > fun_fu8422113094057456412t_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_int_nat > fun_fu5706916984221235812t_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_complex_nat > fun_fu5450758583235971936x_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_nat_real ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_nat_bool > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: assn > fun_fu2678428474534703935l_assn ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: assn > fun_fu1723287682096855374l_assn ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: assn > fun_fu8734043002210220995t_assn ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: assn > fun_fu8831737957336725536t_assn ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: assn > fun_fu96848488177702211t_assn ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: assn > fun_fu1092876921774764064t_assn ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: assn > fun_fu8979060164380311043i_assn ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: assn > fun_fu8850137702660867424i_assn ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: vEBT_VEBT > fun_bool_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_bo3288154511880838028t_bool ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_real > fun_fu6483983356097304844l_bool ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_real > fun_fu415297021002846548l_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_VEBT_VEBT > fun_fu622286109622354600T_bool ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_VEBT_VEBT > fun_fu2956067580704323004T_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_nat > fun_fu6838017323210031556t_bool ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_nat > fun_fu3910677444521609704t_bool ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_int > fun_fu420584378438320396t_bool ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_int > fun_fu656925638523502036t_bool ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_complex > fun_fu7594865232509978120x_bool ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_complex > fun_fu3476074416393937614x_bool ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: set_VEBT_VEBT > fun_fu8277615433164260398T_bool ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_VEBT_VEBT > fun_fu1247336914219032636T_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_VEBT_VEBT > fun_fu4158003351038161580T_bool ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_VEBT_VEBT > fun_fu5001625740209071164T_bool ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_nat_nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_int_int > fun_int_complex ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: vEBT_VEBT > fun_fu8659448316070137091omplex ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: set_VEBT_VEBT > fun_fu5250730037759307465T_bool ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_VE5569070535936029429omplex > fun_fu6701721618212012456omplex ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_VEBT_VEBT_real > fun_fu5928985204125839270T_real ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_VEBT_VEBT_rat > fun_fu8873448384659826898BT_rat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_complex > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_complex_nat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_complex_int > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_int_Code_integer > fun_int_bool ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_co8153928105989009288nteger > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: vEBT_VEBT > fun_fu7059288016784409353BT_int ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: int > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_VE9219535225822561060nteger > fun_fu3480337415675744067nteger ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_VEBT_VEBT_rat > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_Co8310303586393106575r_bool ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: code_integer > fun_na8856381646659805649nteger ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: array_VEBT_VEBTi > fun_li7822782016457402668i_assn ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: real > fun_fu2743955535892867401t_real ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_real_bool ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_real_bool ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_int_bool ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_real_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_real_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_real ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_real_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_real_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_bool_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_bool ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Array__Time_Ofreeze_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_8141364883501958055_VEBTi: array_VEBT_VEBTi > heap_T4980287057938770641_VEBTi ).

tff(sy_c_Array__Time_Onth_001t__HOL__Obool,type,
    array_nth_bool: ( array_bool * nat ) > heap_Time_Heap_bool ).

tff(sy_c_Array__Time_Onth_001t__Int__Oint,type,
    array_nth_int: ( array_int * nat ) > heap_Time_Heap_int ).

tff(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: ( array_nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    array_nth_option_nat: ( array_option_nat * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_nth_VEBT_VEBTi: ( array_VEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: ( assn * produc3658429121746597890et_nat ) > $o ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( assn * assn ) > $o ).

tff(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: fun_li3352006551604695530i_assn > assn ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_bool_assn ).

tff(sy_c_Assertions_Osnga__assn_001t__HOL__Obool,type,
    snga_assn_bool: ( array_bool * list_bool ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Int__Oint,type,
    snga_assn_int: ( array_int * list_int ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: ( array_nat * list_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: ( array_option_nat * list_option_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    snga_assn_VEBT_VEBTi: ( array_VEBT_VEBTi * list_VEBT_VEBTi ) > assn ).

tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: ( assn * assn * assn ) > $o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
    bit_bi6516823479961619367ts_int: fun_nat_bool > int ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun_nat_bool > $o ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option_num ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_bool ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option_num ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
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tff(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: ( code_integer * bool ) > code_integer ).

tff(sy_c_Bits__Integer_Obin__last__integer,type,
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tff(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: code_integer > code_integer ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odup,type,
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tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
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tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: fun_complex_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > fun_real_complex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: fun_complex_complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( fun_real_real * filter_real ) > $o ).

tff(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( fun_real_real * fun_real_real * filter_real ) > $o ).

tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( fun_real_real * real * filter_real ) > $o ).

tff(sy_c_Divides_Oadjust__div,type,
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tff(sy_c_Divides_Odivmod__nat,type,
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tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
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tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__HOL__Obool_001t__Int__Oint,type,
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tff(sy_c_HOL_OThe_001t__Real__Oreal,type,
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tff(sy_c_Heap__Time__Monad_Oreturn_001t__HOL__Obool,type,
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tff(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
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tff(sy_c_Heap__Time__Monad_Oreturn_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001t__HOL__Obool,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__HOL__Obool_J,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
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tff(sy_c_If_001t__Complex__Ocomplex,type,
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tff(sy_c_If_001t__Int__Oint,type,
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tff(sy_c_If_001t__Nat__Onat,type,
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tff(sy_c_If_001t__Num__Onum,type,
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tff(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_If_001t__Rat__Orat,type,
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tff(sy_c_If_001t__Real__Oreal,type,
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tff(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Int_Oint__ge__less__than,type,
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tff(sy_c_Int_Oint__ge__less__than2,type,
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tff(sy_c_Int_Onat,type,
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tff(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
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tff(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
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tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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tff(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
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tff(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_List_Oappend_001t__Int__Oint,type,
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tff(sy_c_List_Oappend_001t__Nat__Onat,type,
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tff(sy_c_List_Odistinct_001t__Int__Oint,type,
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tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: fun_list_nat_bool ).

tff(sy_c_List_Ofilter_001t__Nat__Onat,type,
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tff(sy_c_List_Ofoldr_001t__HOL__Obool_001t__Nat__Onat,type,
    foldr_bool_nat: ( fun_bool_fun_nat_nat * list_bool ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
    foldr_real_nat: ( fun_real_fun_nat_nat * list_real ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( fun_re7681423839302404990l_real * list_real ) > fun_real_real ).

tff(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_OCons_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_ONil_001t__HOL__Obool,type,
    nil_bool: list_bool ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    map_complex_complex: ( fun_complex_complex * list_complex ) > list_complex ).

tff(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    map_complex_real: ( fun_complex_real * list_complex ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__HOL__Obool,type,
    map_bool_bool: ( fun_bool_bool * list_bool ) > list_bool ).

tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__Nat__Onat,type,
    map_bool_nat: ( fun_bool_nat * list_bool ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__Real__Oreal,type,
    map_bool_real: ( fun_bool_real * list_bool ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
    map_int_real: ( fun_int_real * list_int ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__HOL__Obool,type,
    map_nat_bool: ( fun_nat_bool * list_nat ) > list_bool ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( fun_nat_nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE7029150624388687525_VEBTi: ( fun_VE712148701811721852_VEBTi * list_VEBT_VEBT ) > list_VEBT_VEBTi ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( fun_VE3165209603783583043T_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

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

tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
    set_bool2: list_bool > set_bool ).

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

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

tff(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > nat ).

tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

tff(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
    list_update_complex: ( list_complex * nat * complex ) > list_complex ).

tff(sy_c_List_Olist__update_001t__HOL__Obool,type,
    list_update_bool: ( list_bool * nat * bool ) > list_bool ).

tff(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: ( list_int * nat * int ) > list_int ).

tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

tff(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Onth_001t__HOL__Obool,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > fun_nat_int ).

tff(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > fun_nat_nat ).

tff(sy_c_List_Onth_001t__Num__Onum,type,
    nth_num: list_num > fun_nat_num ).

tff(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    nth_option_nat: ( list_option_nat * nat ) > option_nat ).

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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr6329974346453275474_VEBTi: ( list_P785718909624839377_VEBTi * nat ) > produc3777764054643897931_VEBTi ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: ( list_P2623026923184700063T_real * nat ) > produc5170161368751668367T_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > fun_nat_real ).

tff(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: ( list_VEBT_VEBTi * nat ) > vEBT_VEBTi ).

tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: ( list_VEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Int__Oint,type,
    product_bool_int: ( list_bool * list_int ) > list_P6751438739536979202ol_int ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Nat__Onat,type,
    product_bool_nat: ( list_bool * list_nat ) > list_P18149847613930150ol_nat ).

tff(sy_c_List_Oproduct_001t__HOL__Obool_001t__Real__Oreal,type,
    product_bool_real: ( list_bool * list_real ) > list_P8975979458798387586l_real ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__HOL__Obool,type,
    product_nat_bool: ( list_nat * list_bool ) > list_P6781292141273955916t_bool ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Real__Oreal,type,
    product_nat_real: ( list_nat * list_real ) > list_P3644420460460130531t_real ).

tff(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: ( list_num * list_num ) > list_P3744719386663036955um_num ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__HOL__Obool,type,
    product_real_bool: ( list_real * list_bool ) > list_P3020538015149218416l_bool ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Int__Oint,type,
    product_real_int: ( list_real * list_int ) > list_P4344331454722006975al_int ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: ( list_real * list_nat ) > list_P6834414599653733731al_nat ).

tff(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: ( list_real * list_real ) > list_P8689742595348180415l_real ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__HOL__Obool,type,
    produc9101113540930457115i_bool: ( list_VEBT_VEBTi * list_bool ) > list_P4809559687441107495i_bool ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc5476717833281694120i_real: ( list_VEBT_VEBTi * list_real ) > list_P8536626330812492744i_real ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc194614972289024177_VEBTi: ( list_VEBT_VEBTi * list_VEBT_VEBTi ) > list_P785718909624839377_VEBTi ).

tff(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc1285381384045549624T_VEBT: ( list_VEBT_VEBTi * list_VEBT_VEBT ) > list_P5988454224134618948T_VEBT ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    produc5466675335756156456T_bool: ( list_VEBT_VEBT * list_bool ) > list_P44034876932920720T_bool ).

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

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc4908677263432625371T_real: ( list_VEBT_VEBT * list_real ) > list_P2623026923184700063T_real ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

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

tff(sy_c_List_Oreplicate_001t__HOL__Obool,type,
    replicate_bool: ( nat * bool ) > list_bool ).

tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: ( nat * int ) > list_int ).

tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: ( nat * real ) > list_real ).

tff(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    replicate_VEBT_VEBTi: ( nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

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

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr3099820731930151597t_bool ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__HOL__Obool,type,
    case_nat_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__HOL__Obool_J,type,
    size_size_list_bool: list_bool > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
    size_size_list_num: list_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6086282163384603972on_nat: list_option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__HOL__Obool_J_J,type,
    size_s6988337448923680857l_bool: list_P2722043638750504429l_bool > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__HOL__Obool_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s7982070591426661849_VEBTi: list_VEBT_VEBTi > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: ( option_num * fun_num_option_num * fun_num_option_num * num ) > option_num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

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

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

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

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

tff(sy_c_Num_Opow,type,
    pow: fun_num_fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    none_P533106815845188193et_nat: option936205604648967762et_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Code____Numeral__Ointeger,type,
    some_Code_integer: code_integer > option_Code_integer ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

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

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

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(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 ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

tff(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: ( int * fun_num_int * option_num ) > int ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: ( num * fun_num_num * option_num ) > num ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: ( option_num * fun_num_option_num * option_num ) > option_num ).

tff(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( fun_nat_nat * option_nat ) > nat ).

tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
    size_option_num: ( fun_num_nat * option_num ) > nat ).

tff(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    size_o8335143837870341156at_nat: ( fun_Pr2602117396471556196at_nat * option4927543243414619207at_nat ) > nat ).

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

tff(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: option_num > num ).

tff(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(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 ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: ( option_real * option_real ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__String__Ochar,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > fun_set_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > fun_set_real_bool ).

tff(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 ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: ( set_VEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(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 ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
    ord_less_eq_char: ( char * char ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: ( rat * rat ) > rat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HOL__Obool_J,type,
    top_top_set_bool: set_bool ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
    top_top_set_literal: set_literal ).

tff(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > fun_nat_assn ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > fun_nat_complex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: ( heap_e7401611519738050253t_unit * set_nat ) > produc3658429121746597890et_nat ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: ( int * int ) > product_prod_int_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: ( nat * num ) > product_prod_nat_num ).

tff(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 ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc1195630363706982562at_num: ( nat * product_prod_nat_num ) > produc2963631642982155120at_num ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: ( option_nat * option_nat ) > produc4953844613479565601on_nat ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    produc8585076106096196333on_num: ( option_num * option_num ) > produc3447558737645232053on_num ).

tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: ( option4927543243414619207at_nat * option4927543243414619207at_nat ) > produc6121120109295599847at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__HOL__Obool,type,
    produc7282046070707253577i_bool: ( vEBT_VEBTi * bool ) > produc4444102050130342551i_bool ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc8457151488442208762i_real: ( vEBT_VEBTi * real ) > produc6680258955013199682i_real ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc436343169921013763_VEBTi: ( vEBT_VEBTi * vEBT_VEBTi ) > produc3777764054643897931_VEBTi ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc7053807326796202854T_VEBT: ( vEBT_VEBTi * vEBT_VEBT ) > produc2810682830582626868T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc6084888613844515218_VEBTi: ( vEBT_VEBT * vEBT_VEBTi ) > produc3625547720036274456_VEBTi ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_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,
    produc2818902965508115497at_nat: ( fun_nat_fun_nat_nat * produc7248412053542808358at_nat ) > produc9217583074051984497at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc6699827698519632094on_nat: ( fun_nat_fun_nat_nat * produc4953844613479565601on_nat ) > produc7004362771917097900on_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__HOL__Obool_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: int > fun_int_int ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > fun_nat_bool ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: ( assn * heap_Time_Heap_nat * fun_nat_assn ) > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: ( assn * heap_T2636463487746394924on_nat * fun_option_nat_assn ) > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: ( assn * heap_T8145700208782473153_VEBTi * fun_VEBT_VEBTi_assn ) > fun_nat_bool ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__HOL__Obool,type,
    vEBT_V6044891547498605301i_bool: ( nat * heap_Time_Heap_bool ) > heap_T5178755288483056679t_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: ( nat * heap_Time_Heap_nat ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: ( nat * heap_T2636463487746394924on_nat ) > heap_T5317711798761887292on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: ( nat * heap_T8145700208782473153_VEBTi ) > heap_T4980287057938770641_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap_bool ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( bool * bool ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option4927543243414619207at_nat * nat * array_VEBT_VEBTi ) > fun_VE8449866691444886503_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( fun_op2425996437410981688ap_nat * fun_bo5340758216624605938ap_nat * vEBT_VEBTi ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( fun_op5783775701855621896on_nat * fun_bo8430231047727935170on_nat * vEBT_VEBTi ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( fun_op5127434927081926045_VEBTi * fun_bo851793448214054487_VEBTi * vEBT_VEBTi ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( fun_op7256438278201533260Ti_nat * fun_bo5748384255075588070ol_nat ) > fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr9085508990737135641i_bool ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun_VE6376819126223186655i_bool ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun_VE6376819126223186655i_bool ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__HOL__Obool_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6372884366951783724_VEBTi: ( set_nat * fun_bo7535889243153246212i_assn * list_bool * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2992182275864018813T_VEBT: ( set_nat * fun_bo5394993412183903963T_assn * list_bool * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2018189785592951398T_VEBT: ( set_nat * fun_in7755481285845205148T_assn * list_int * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7489483478785760935_VEBTi: ( set_nat * fun_na6467670216208870567i_assn * list_nat * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8511957252848910786T_VEBT: ( set_nat * fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: ( set_nat * fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: ( set_nat * fun_re560840401387337987i_assn * list_real * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: ( set_nat * fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__HOL__Obool,type,
    vEBT_L2302322349616425472i_bool: ( set_nat * fun_VE3808733824349961398l_assn * list_VEBT_VEBTi * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L2806540629473551875Ti_int: ( set_nat * fun_VE5641251438918192003t_assn * list_VEBT_VEBTi * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L2809031099982602151Ti_nat: ( set_nat * fun_VE1225457145571040039t_assn * list_VEBT_VEBTi * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L7728200936804140803i_real: ( set_nat * fun_VE1885522831824361987l_assn * list_VEBT_VEBTi * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L886525131989349516_VEBTi: ( set_nat * fun_VE2561185598974572172i_assn * list_VEBT_VEBTi * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2497118539674116125T_VEBT: ( set_nat * fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    vEBT_L6176004506706552451T_bool: ( set_nat * fun_VE1323945509527881313l_assn * list_VEBT_VEBT * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8648204552663881920BT_int: ( set_nat * fun_VE123837653852366616t_assn * list_VEBT_VEBT * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: ( set_nat * fun_VE4931415397359990460t_assn * list_VEBT_VEBT * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: ( set_nat * fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: ( set_nat * fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    vEBT_L4260503343685368993omplex: ( fun_co6163879640491479806x_assn * list_complex * list_complex ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Int__Oint,type,
    vEBT_L134985006839036959ex_int: ( fun_co6164383388265235324t_assn * list_complex * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    vEBT_L137475477348087235ex_nat: ( fun_co1748589094918083360t_assn * list_complex * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    vEBT_L2479436891206192927x_real: ( fun_co8000059489268409724l_assn * list_complex * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8524933119956041985T_VEBT: ( fun_co2031555705788908698T_assn * list_complex * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__HOL__Obool,type,
    vEBT_L6562251693209944067l_bool: ( fun_bo8172996160782635326l_assn * list_bool * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Nat__Onat,type,
    vEBT_L7520539408223738084ol_nat: ( fun_bo8438563643002758303t_assn * list_bool * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__HOL__Obool_001t__Real__Oreal,type,
    vEBT_L5107238702033023424l_real: ( fun_bo8885878543415060347l_assn * list_bool * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__HOL__Obool,type,
    vEBT_L8658821491502452966t_bool: ( fun_in4945713774130038527l_assn * list_int * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( fun_in6446958447386763770l_assn * list_int * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__HOL__Obool,type,
    vEBT_L2559740421116301378t_bool: ( fun_na5136419689362302555l_assn * list_nat * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( fun_na448437756743965086l_assn * list_nat * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__HOL__Obool,type,
    vEBT_L870199270341830246l_bool: ( fun_re2206215847901373695l_assn * list_real * list_bool ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( fun_re560840401387337987i_assn * list_real * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L8927591528087875366Ti_int: ( fun_VE5641251438918192003t_assn * list_VEBT_VEBTi * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L8930081998596925642Ti_nat: ( fun_VE1225457145571040039t_assn * list_VEBT_VEBTi * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1891944875198410415_VEBTi: ( fun_VE2561185598974572172i_assn * list_VEBT_VEBTi * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    vEBT_L2162147798726695391omplex: ( fun_VE496882281059548826x_assn * list_VEBT_VEBT * list_complex ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool,type,
    vEBT_L5320321625355440422T_bool: ( fun_VE1323945509527881313l_assn * list_VEBT_VEBT ) > fun_list_bool_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( fun_VE123837653852366616t_assn * list_VEBT_VEBT * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( fun_VE4931415397359990460t_assn * list_VEBT_VEBT ) > fun_list_nat_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( fun_VE6334927249922962060t_assn * list_VEBT_VEBT ) > fun_li338670087962420949t_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE3763956218935105199t_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > fun_nat_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( fun_num_fun_num_num * option_num * option_num ) > option_num ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE5877677516038742509T_bool ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
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tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_J,type,
    aa_fun2693410532875676774BT_rat: ( fun_fu4463953228700081077BT_rat * fun_VEBT_VEBT_rat ) > fun_fu2188483506383969923BT_rat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J,type,
    aa_fun5605965666826184925T_bool: ( fun_fu4158003351038161580T_bool * fun_VEBT_VEBT_real ) > fun_VEBT_VEBT_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    aa_fun1825138017709814396T_real: ( fun_fu8198533418994508931T_real * fun_VEBT_VEBT_real ) > fun_VEBT_VEBT_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__HOL__Obool_J_J,type,
    aa_fun4082248197810269365T_bool: ( fun_fu5001625740209071164T_bool * fun_VEBT_VEBT_real ) > fun_fu4158003351038161580T_bool ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    aa_fun910392650362481850T_real: ( fun_fu4141963935195952137T_real * fun_VEBT_VEBT_real ) > fun_fu8198533418994508931T_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J,type,
    aa_fun6755658226996116224l_assn: ( fun_fu2678428474534703935l_assn * fun_VE1323945509527881313l_assn ) > fun_VE1323945509527881313l_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__HOL__Obool_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__HOL__Obool_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun7303525306808986255l_assn: ( fun_fu1723287682096855374l_assn * fun_VE1323945509527881313l_assn ) > fun_VE4045106055134628720l_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Assertions__Oassn_J_J,type,
    aa_fun7634423771582967228t_assn: ( fun_fu96848488177702211t_assn * fun_VE4931415397359990460t_assn ) > fun_VE4931415397359990460t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun6018615163312975129t_assn: ( fun_fu1092876921774764064t_assn * fun_VE4931415397359990460t_assn ) > fun_VE7969100967704966681t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun7129419364150962969t_assn: ( fun_fu8831737957336725536t_assn * fun_VE6334927249922962060t_assn ) > fun_VE2615472500133170025t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Assertions__Oassn_J_J,type,
    aa_fun1240582285364316732t_assn: ( fun_fu8734043002210220995t_assn * fun_VE6334927249922962060t_assn ) > fun_VE6334927249922962060t_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__Assertions__Oassn_J_J_J,type,
    aa_fun8635824850342594777i_assn: ( fun_fu8850137702660867424i_assn * fun_VE905575822219544865i_assn ) > fun_VE2880048358281379966i_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J_J,type,
    aa_fun4659777589554175228i_assn: ( fun_fu8979060164380311043i_assn * fun_VE905575822219544865i_assn ) > fun_VE905575822219544865i_assn ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_bool > real ).

tff(sy_c_fEx_001t__Real__Oreal,type,
    fEx_real: fun_fu7511176317168184086l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: bool > fun_bool_bool ).

tff(sy_c_fdisj,type,
    fdisj: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
    fequal_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1378794544173401837x_bool ).

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    fequal1389750275112723248_VEBTi: ( list_VEBT_VEBTi * list_VEBT_VEBTi ) > bool ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Nat__Onat_J,type,
    fequal_option_nat: fun_op7792301606653096457t_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Num__Onum_J,type,
    fequal_option_num: fun_op1962577433978643573m_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fequal3124746952608472732at_nat: fun_op5682630191948829869t_bool ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    fequal6010348034681699726nt_int: product_prod_int_int > fun_Pr803113274115125473t_bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fequal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    fequal_VEBT_VEBTi: vEBT_VEBTi > fun_VEBT_VEBTi_bool ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: vEBT_VEBT > fun_VEBT_VEBT_bool ).

tff(sy_c_fimplies,type,
    fimplies: bool > fun_bool_bool ).

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > fun_se2712834123562403997r_bool ).

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

tff(sy_c_member_001t__HOL__Obool,type,
    member_bool: ( bool * set_bool ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: int > fun_set_int_bool ).

tff(sy_c_member_001t__List__Olist_It__HOL__Obool_J,type,
    member_list_bool: ( list_bool * set_list_bool ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: ( list_real * set_list_real ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_bool ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: num > fun_set_num_bool ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: ( produc3658429121746597890et_nat * set_Pr3948176798113811640et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(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 ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > fun_set_rat_bool ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: real > fun_set_real_bool ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

tff(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: ( vEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_tree__is__103_058ATP,type,
    tree_is_103_ATP: list_VEBT_VEBTi ).

tff(sy_v_tree__is______,type,
    tree_is: list_VEBT_VEBTi ).

tff(sy_v_va____,type,
    va: nat ).

tff(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

tff(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10737)
tff(fact_0_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14) = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_1_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is)),aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ).

% assnle
tff(fact_2_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_3_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_4_zero__less__power2,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_5_zero__less__power2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_6_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% div2_Suc_Suc
tff(fact_7_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_8_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_9_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_10_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_11_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_12_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_13_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_14_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_15_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_16_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_17_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_18_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_19_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_20_power__eq__0__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_21_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_22_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_23_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_24_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_25_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_26_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
    <=> ( ( ( numeral_numeral_real(W) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) ) )
        & ( ( numeral_numeral_real(W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_27_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> ( ( ( numeral_numeral_rat(W) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) ) )
        & ( ( numeral_numeral_rat(W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_28_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B ) )
        & ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_29_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
    <=> ( ( ( numeral_numeral_real(W) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B ) )
        & ( ( numeral_numeral_real(W) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_30_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> ( ( ( numeral_numeral_rat(W) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) = B ) )
        & ( ( numeral_numeral_rat(W) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_31_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_32_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_33_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_34_odd__power__less__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),zero_z3403309356797280102nteger)) ) ).

% odd_power_less_zero
tff(fact_35_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_36_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_37_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_38_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_39_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_40_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_41_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,bit0(M)),bit0(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(78)
tff(fact_42_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,bit0(M)),bit0(N)) = bit0(bit0(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% semiring_norm(13)
tff(fact_43_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0(M) = bit0(N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_44_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).

% semiring_norm(75)
tff(fact_45_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).

% semiring_norm(11)
tff(fact_46_semiring__norm_I12_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),N) = N ).

% semiring_norm(12)
tff(fact_47_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_48_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_49_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_50_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_51_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% numeral_less_iff
tff(fact_52_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_53_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_54_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_55_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_56_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_57_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_58_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_59_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_60_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_61_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_62_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),bit0(N))) ).

% semiring_norm(76)
tff(fact_63_num__double,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,bit0(one)),N) = bit0(N) ).

% num_double
tff(fact_64_semiring__norm_I83_J,axiom,
    ! [N: num] : one != bit0(N) ).

% semiring_norm(83)
tff(fact_65_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one ).

% semiring_norm(85)
tff(fact_66_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_67_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_68_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_69_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_70_power__mult__numeral,axiom,
    ! [A: code_integer,M: num,N: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_71_power__mult__numeral,axiom,
    ! [A: rat,M: num,N: num] : aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_72_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_mult_mult1_if
tff(fact_73_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_74_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_75_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult2
tff(fact_76_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_77_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_78_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult1
tff(fact_79_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_80_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_81_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_82_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_83_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_84_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_85_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,N)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_86_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_87_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_88_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_89_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_90_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_91_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_92_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_93_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_94_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_95_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_96_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_97_power__Suc0__right,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_98_power__Suc0__right,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_99_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_100_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_101_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_102_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_103_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_104_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_105_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_106_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_107_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(member_real(A),collect_real(P)))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_108_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( member_VEBT_VEBT(A,collect_VEBT_VEBT(P))
    <=> pp(aa_VEBT_VEBT_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_109_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_110_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P)))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_111_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_bool] :
      ( member_complex(A,collect_complex(P))
    <=> pp(aa_complex_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_112_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: fun_Pr803113274115125473t_bool] :
      ( member5262025264175285858nt_int(A,collec213857154873943460nt_int(P))
    <=> pp(aa_Pro4501881432599481050t_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_113_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_114_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_115_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_116_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_117_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_118_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_119_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_nat_bool(P,X2))
        <=> pp(aa_nat_bool(Q,X2)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_120_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(Q,X2)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_121_Collect__cong,axiom,
    ! [P: fun_complex_bool,Q: fun_complex_bool] :
      ( ! [X2: complex] :
          ( pp(aa_complex_bool(P,X2))
        <=> pp(aa_complex_bool(Q,X2)) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_122_Collect__cong,axiom,
    ! [P: fun_Pr803113274115125473t_bool,Q: fun_Pr803113274115125473t_bool] :
      ( ! [X2: product_prod_int_int] :
          ( pp(aa_Pro4501881432599481050t_bool(P,X2))
        <=> pp(aa_Pro4501881432599481050t_bool(Q,X2)) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_123_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_124_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_125_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_126_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_127_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_numeral_extra(3)
tff(fact_128_zero__neq__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != numera6620942414471956472nteger(N) ).

% zero_neq_numeral
tff(fact_129_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).

% zero_neq_numeral
tff(fact_130_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != numeral_numeral_real(N) ).

% zero_neq_numeral
tff(fact_131_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != numeral_numeral_rat(N) ).

% zero_neq_numeral
tff(fact_132_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_133_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_134_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_135_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_136_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_137_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_138_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_139_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_140_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_141_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),Y) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),X) )
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),Y) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_142_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_143_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y) = aa_rat_rat(times_times_rat(Y),X) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_144_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_145_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(times_times_int(X),Y) = aa_int_int(times_times_int(Y),X) )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(power_power_int(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_146_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).

% power_mult_distrib
tff(fact_147_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),N) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ).

% power_mult_distrib
tff(fact_148_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)) ).

% power_mult_distrib
tff(fact_149_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).

% power_mult_distrib
tff(fact_150_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).

% power_mult_distrib
tff(fact_151_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(times_times_int(A),B)),N) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).

% power_mult_distrib
tff(fact_152_power__commutes,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_commutes
tff(fact_153_power__commutes,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_commutes
tff(fact_154_power__commutes,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_commutes
tff(fact_155_power__commutes,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).

% power_commutes
tff(fact_156_power__commutes,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_commutes
tff(fact_157_power__commutes,axiom,
    ! [A: int,N: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),N)),A) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N)) ).

% power_commutes
tff(fact_158_power__divide,axiom,
    ! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),N) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).

% power_divide
tff(fact_159_power__divide,axiom,
    ! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)) ).

% power_divide
tff(fact_160_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),N) = divide_divide_rat(aa_nat_rat(power_power_rat(A),N),aa_nat_rat(power_power_rat(B),N)) ).

% power_divide
tff(fact_161_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_162_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),N) ).

% power_mult
tff(fact_163_power__mult,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),N) ).

% power_mult
tff(fact_164_power__mult,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),N) ).

% power_mult
tff(fact_165_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),N) ).

% power_mult
tff(fact_166_power__mult,axiom,
    ! [A: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),N) ).

% power_mult
tff(fact_167_power__mult,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),M)),N) ).

% power_mult
tff(fact_168_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2) ).

% div_mult2_eq
tff(fact_169_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),zero_z3403309356797280102nteger)) ).

% not_numeral_less_zero
tff(fact_170_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_171_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_172_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_173_not__numeral__less__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_174_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(N))) ).

% zero_less_numeral
tff(fact_175_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(N))) ).

% zero_less_numeral
tff(fact_176_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),numeral_numeral_rat(N))) ).

% zero_less_numeral
tff(fact_177_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_less_numeral
tff(fact_178_zero__less__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_less_numeral
tff(fact_179_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_less_power
tff(fact_180_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_less_power
tff(fact_181_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_less_power
tff(fact_182_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_less_power
tff(fact_183_zero__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% zero_less_power
tff(fact_184_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_185_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).

% mult_numeral_1_right
tff(fact_186_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_187_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_188_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_189_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_190_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).

% mult_numeral_1
tff(fact_191_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_192_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_193_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_194_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_195_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).

% divide_numeral_1
tff(fact_196_divide__numeral__1,axiom,
    ! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).

% divide_numeral_1
tff(fact_197_power__Suc2,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) ).

% power_Suc2
tff(fact_198_power__Suc2,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),A) ).

% power_Suc2
tff(fact_199_power__Suc2,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) ).

% power_Suc2
tff(fact_200_power__Suc2,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) ).

% power_Suc2
tff(fact_201_power__Suc2,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).

% power_Suc2
tff(fact_202_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_203_power__Suc,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).

% power_Suc
tff(fact_204_power__Suc,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_Suc
tff(fact_205_power__Suc,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_Suc
tff(fact_206_power__Suc,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).

% power_Suc
tff(fact_207_power__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_208_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_209_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_210_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_211_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_212_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_213_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_214_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_215_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( numeral_numeral_real(W) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_216_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( numeral_numeral_rat(W) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_217_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_218_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) ) )
        & ( ( C = zero_zero_real )
         => ( numeral_numeral_real(W) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_219_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( numeral_numeral_rat(W) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_220_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,bit0(N))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit0_div_2
tff(fact_221_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,bit0(N))),aa_num_int(numeral_numeral_int,bit0(one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit0_div_2
tff(fact_222_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_223_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_224_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_225_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_226_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_227_zero__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_228_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_229_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).

% power_gt_expt
tff(fact_230_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_231_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_232_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_233_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_234_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_235_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_236_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% zero_power2
tff(fact_237_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_real ).

% zero_power2
tff(fact_238_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_int ).

% zero_power2
tff(fact_239_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_complex ).

% zero_power2
tff(fact_240_zero__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_241_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_rat ).

% zero_power2
tff(fact_242_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_243_power2__eq__square,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A) ).

% power2_eq_square
tff(fact_244_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_245_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_rat_rat(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_246_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_247_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_int_int(times_times_int(A),A) ).

% power2_eq_square
tff(fact_248_power4__eq__xxxx,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_249_power4__eq__xxxx,axiom,
    ! [X: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_250_power4__eq__xxxx,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_251_power4__eq__xxxx,axiom,
    ! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_252_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_253_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(times_times_int(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_254_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,bit0(one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_255_power__even__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power_even_eq
tff(fact_256_power__even__eq,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power_even_eq
tff(fact_257_power__even__eq,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power_even_eq
tff(fact_258_power__even__eq,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power_even_eq
tff(fact_259_power__even__eq,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power_even_eq
tff(fact_260_power__even__eq,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power_even_eq
tff(fact_261_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(bit0(one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_262_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(bit0(one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% half_gt_zero_iff
tff(fact_263_half__gt__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(bit0(one))))) ) ).

% half_gt_zero
tff(fact_264_half__gt__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(bit0(one))))) ) ).

% half_gt_zero
tff(fact_265_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_266_power2__less__0,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_267_power2__less__0,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_268_power2__less__0,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_z3403309356797280102nteger)) ).

% power2_less_0
tff(fact_269_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_270_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_271_power__odd__eq,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_272_power__odd__eq,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_273_power__odd__eq,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_274_power__odd__eq,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_275_power__odd__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_276_power__odd__eq,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% power_odd_eq
tff(fact_277_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_278_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% div_2_gt_zero
tff(fact_279_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_280_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_281_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_282_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_283_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_284_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_285_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_286_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_287_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_288_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_289_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_290_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_291_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_292_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_293_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_294_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_295_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_296_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_297_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_298_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_299_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_300_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_301_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_302_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_303_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_304_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_305_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_306_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_307_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_308_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_309_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_310_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_311_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_312_mult__zero__left,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_313_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_314_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_315_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_316_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(times_times_int(zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_317_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_318_mult__zero__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_319_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_320_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_321_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_322_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_323_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_324_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_325_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_326_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_327_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_328_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_329_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_330_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_331_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_332_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_333_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_334_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_335_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_336_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_337_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_338_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_339_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_340_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_341_div__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% div_0
tff(fact_342_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_343_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_344_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_345_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_346_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_347_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_348_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_349_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
tff(fact_350_div__by__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% div_by_0
tff(fact_351_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_352_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_353_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_354_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_355_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_356_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_357_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_358_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
tff(fact_359_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_360_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_361_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
tff(fact_362_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_363_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_364_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_365_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_366_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_367_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(times_times_rat(B),A),C) ).

% times_divide_eq_left
tff(fact_368_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_369_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_370_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_371_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_372_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_373_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),C),B) ).

% divide_divide_eq_right
tff(fact_374_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_375_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_376_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),B),C) ).

% times_divide_eq_right
tff(fact_377_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_378_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_379_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_eq
tff(fact_380_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_381_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_382_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_383_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_384_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_385_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_386_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_387_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_388_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_389_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_390_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X3)) ).

% linordered_field_no_lb
tff(fact_391_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X3)) ).

% linordered_field_no_lb
tff(fact_392_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X_1)) ).

% linordered_field_no_ub
tff(fact_393_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),X_1)) ).

% linordered_field_no_ub
tff(fact_394_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_395_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_396_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_397_linorder__neqE__linordered__idom,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_398_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_399_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_400_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_401_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_402_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_403_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_404_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_405_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_406_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa_nat_bool(P,N2))
         => ? [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
              & ~ pp(aa_nat_bool(P,M2)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_407_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_408_size__neq__size__imp__neq,axiom,
    ! [X: list_real,Y: list_real] :
      ( ( size_size_list_real(X) != size_size_list_real(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_409_size__neq__size__imp__neq,axiom,
    ! [X: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X) != size_size_list_bool(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_410_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
tff(fact_411_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_412_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_413_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_414_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_415_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_416_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_417_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_418_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_419_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_420_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_421_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_422_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_423_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_424_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_425_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_426_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_427_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_428_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_429_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_430_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(times_times_int(A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_431_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_432_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_433_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_434_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_435_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_436_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_437_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_438_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_439_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_440_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_441_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_442_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_443_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_444_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_445_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(X,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_446_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_447_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_448_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(X,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),W),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_449_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_450_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_451_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,aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_452_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_453_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_454_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_455_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_456_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_457_nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_induct
tff(fact_458_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X2: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).

% diff_induct
tff(fact_459_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_460_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_461_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_462_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_463_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_464_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
             => ( K != aa_nat_nat(suc,J) ) ) ) ) ).

% Nat.lessE
tff(fact_465_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_lessD
tff(fact_466_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
           => ( K != aa_nat_nat(suc,J) ) ) ) ).

% Suc_lessE
tff(fact_467_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( ( aa_nat_nat(suc,M) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_468_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_469_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_470_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I2)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
            & pp(aa_nat_bool(P,I2)) ) ) ) ).

% Ex_less_Suc
tff(fact_471_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_472_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_473_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I2)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
           => pp(aa_nat_bool(P,I2)) ) ) ) ).

% Nat.All_less_Suc
tff(fact_474_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
    <=> ? [M4: nat] :
          ( ( M = aa_nat_nat(suc,M4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M4)) ) ) ).

% Suc_less_eq2
tff(fact_475_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_476_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_477_less__trans__Suc,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_478_less__Suc__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( ! [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(suc,I3)))
       => ( ! [I3: nat,J: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),J))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J2)) ) ) ) ).

% less_Suc_induct
tff(fact_479_strict__inc__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( ! [I3: nat] :
            ( ( J2 = aa_nat_nat(suc,I3) )
           => pp(aa_nat_bool(P,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J2))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I3)))
               => pp(aa_nat_bool(P,I3)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_480_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_481_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_482_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_483_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_484_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_485_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_486_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_487_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( ~ pp(aa_nat_bool(P,N2))
             => ? [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
                  & ~ pp(aa_nat_bool(P,M2)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_488_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_489_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_490_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_neg_neg
tff(fact_491_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_neg_neg
tff(fact_492_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))) ) ) ).

% mult_neg_neg
tff(fact_493_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))) ) ) ).

% mult_neg_neg
tff(fact_494_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_495_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_496_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),A)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_497_not__square__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A)),zero_z3403309356797280102nteger)) ).

% not_square_less_zero
tff(fact_498_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_499_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_500_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_501_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_502_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_503_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_504_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_505_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_506_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)) ) ) ).

% mult_neg_pos
tff(fact_507_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_508_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_509_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_510_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_511_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)) ) ) ).

% mult_pos_neg
tff(fact_512_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_pos_pos
tff(fact_513_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_pos_pos
tff(fact_514_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_515_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))) ) ) ).

% mult_pos_pos
tff(fact_516_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))) ) ) ).

% mult_pos_pos
tff(fact_517_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_518_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_519_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_520_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_521_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)),zero_z3403309356797280102nteger)) ) ) ).

% mult_pos_neg2
tff(fact_522_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_523_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_524_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_525_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)) ) ) ) ).

% zero_less_mult_iff
tff(fact_526_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos
tff(fact_527_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_528_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_529_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos
tff(fact_530_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ) ).

% zero_less_mult_pos
tff(fact_531_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_532_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_533_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_534_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(times_times_int(B),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_535_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_536_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_537_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_538_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_539_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_540_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_541_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_542_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_543_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_544_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_545_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_546_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_547_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_548_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_549_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_550_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_551_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_552_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_553_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_554_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_555_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_556_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_557_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_558_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_559_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_560_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_561_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_562_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_563_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_564_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_565_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_566_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_567_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_568_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_569_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_570_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_571_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_572_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_573_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_574_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_575_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_576_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).

% divide_neg_neg
tff(fact_577_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_578_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_579_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_580_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_581_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_582_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).

% divide_pos_pos
tff(fact_583_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_584_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_585_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_586_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_587_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_588_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_589_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_590_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).

% divide_strict_right_mono
tff(fact_591_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_592_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_593_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_594_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_595_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(X,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_596_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_597_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_598_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_599_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_600_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_601_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_602_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_603_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_604_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_605_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_606_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_607_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_608_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_609_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_610_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_611_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_612_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_613_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_614_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_615_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_616_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_617_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_618_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_619_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_Code_integer,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N2)),aa_nat_Code_integer(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_620_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_621_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_622_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_623_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_624_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_625_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_Code_integer,N: nat,M: nat] :
      ( ! [N2: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N2)),aa_nat_Code_integer(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,N)),aa_nat_Code_integer(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_626_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I2)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I2))) ) ) ) ).

% Ex_less_Suc2
tff(fact_627_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M5: nat] : N = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_628_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I2)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I2))) ) ) ) ).

% All_less_Suc2
tff(fact_629_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% gr0_implies_Suc
tff(fact_630_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_631_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_632_mult__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K))) ) ) ).

% mult_less_mono1
tff(fact_633_mult__less__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J2))) ) ) ).

% mult_less_mono2
tff(fact_634_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_635_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_636_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_637_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_638_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_639_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_640_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_641_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_642_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_643_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_644_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_645_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_646_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_647_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),Z)) ) ) ).

% mult_imp_div_pos_less
tff(fact_648_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_649_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),divide_divide_rat(X,Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_650_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_651_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_652_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_653_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_654_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% one_less_mult
tff(fact_655_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_656_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_657_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_658_listI__assn__conv_H,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A2: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi,F2: assn] :
      ( ( N = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_assn_assn(times_times_assn(vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N),A2,Xs,Xsi)),F2) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs),Xsi)),F2) ) ) ).

% listI_assn_conv'
tff(fact_659_listI__assn__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A2: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi] :
      ( ( N = size_s6755466524823107622T_VEBT(Xs) )
     => ( vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N),A2,Xs,Xsi) = aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_660_list__assn__conv__idx,axiom,
    ! [A2: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs),Xsi) = vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),A2,Xs,Xsi) ).

% list_assn_conv_idx
tff(fact_661_n__less__equal__power__2,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% n_less_equal_power_2
tff(fact_662_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(2)
tff(fact_663_less__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ord_less_option_real(some_real(X),some_real(Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% less_option_Some
tff(fact_664_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(X),some_rat(Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% less_option_Some
tff(fact_665_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,some_num(X)),some_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% less_option_Some
tff(fact_666_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,some_nat(X)),some_nat(Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_option_Some
tff(fact_667_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ord_less_option_int(some_int(X),some_int(Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% less_option_Some
tff(fact_668_less__option__Some,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ord_le7113747843092208513nteger(some_Code_integer(X),some_Code_integer(Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).

% less_option_Some
tff(fact_669_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% Suc_double_not_eq_double
tff(fact_670_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% double_not_eq_Suc_double
tff(fact_671_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% pos2
tff(fact_672_bits__div__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_div_0
tff(fact_673_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_674_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_675_bits__div__by__0,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bits_div_by_0
tff(fact_676_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_677_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_678_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_679_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_680_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_681_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_682_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M2),N2))
             => pp(aa_Ext2772979874139974381t_bool(P,M2)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N2)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).

% enat_less_induct
tff(fact_683_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,N)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_684_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R),aa_nat_nat(suc,N)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_685_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_686_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),N) = A ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_687_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [R: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R),N) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_688_list__assn__mono,axiom,
    ! [P: fun_VE905575822219544865i_assn,P2: fun_VE905575822219544865i_assn,L: list_VEBT_VEBT,L2: list_VEBT_VEBTi] :
      ( ! [X2: vEBT_VEBT,X4: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,X2),X4),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P2,X2),X4))
     => entails(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,L),L2),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P2,L),L2)) ) ).

% list_assn_mono
tff(fact_689_four__x__squared,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% four_x_squared
tff(fact_690_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_691_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B) ).

% pow_sum
tff(fact_692_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),B)) ) ) ).

% td_gal_lt
tff(fact_693_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% unset_bit_0
tff(fact_694_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% unset_bit_0
tff(fact_695_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% unset_bit_0
tff(fact_696_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% T_vebt_buildupi_gq_0
tff(fact_697_high__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% high_def
tff(fact_698_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
         => pp(aa_nat_bool(P,M5)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less_eq
tff(fact_699_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
          & pp(aa_nat_bool(P,M5)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less_eq
tff(fact_700_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_701_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(some_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_702_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(some_num(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_703_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,X)))
         => pp(aa_nat_bool(P,I2)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),X))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I2))) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_704_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_705_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_706_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
tff(fact_707_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_708_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_709_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_710_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
tff(fact_711_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_712_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_713_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_714_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( some_nat(X22) = some_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_715_option_Oinject,axiom,
    ! [X22: num,Y2: num] :
      ( ( some_num(X22) = some_num(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_716_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R2) ).

% real_divide_square_eq
tff(fact_717_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ).

% high_bound_aux
tff(fact_718_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),X)),N) = Y ) ) ).

% high_inv
tff(fact_719_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_720_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_721_add_Oright__neutral,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% add.right_neutral
tff(fact_722_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_723_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_724_add_Oright__neutral,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,zero_z3403309356797280102nteger) = A ).

% add.right_neutral
tff(fact_725_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_726_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = plus_plus_rat(A,A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_727_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(plus_plus_int(A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_728_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = plus_p5714425477246183910nteger(A,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
tff(fact_729_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_730_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_731_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( plus_plus_rat(B,A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_732_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_733_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_734_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger(B,A) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
tff(fact_735_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_736_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_737_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_738_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_739_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_740_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
tff(fact_741_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_742_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_743_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = plus_plus_rat(B,A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_744_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_745_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_746_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = plus_p5714425477246183910nteger(B,A) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
tff(fact_747_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_748_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_749_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = plus_plus_rat(A,B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_750_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_751_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_752_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = plus_p5714425477246183910nteger(A,B) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
tff(fact_753_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_754_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_755_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_756_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_757_add__0,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% add_0
tff(fact_758_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_759_add__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add_0
tff(fact_760_add__0,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,A) = A ).

% add_0
tff(fact_761_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_762_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_763_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_764_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_765_add__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(C,A)),plus_p5714425477246183910nteger(C,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% add_less_cancel_left
tff(fact_766_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_767_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_768_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_769_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_770_add__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,C)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% add_less_cancel_right
tff(fact_771_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_772_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_773_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : plus_plus_rat(numeral_numeral_rat(V),plus_plus_rat(numeral_numeral_rat(W),Z)) = plus_plus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_774_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_775_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_776_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_777_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_778_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_779_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_780_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_781_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_782_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_783_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_784_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_785_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_786_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),D))),L) ).

% bit_concat_def
tff(fact_787_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_788_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(B,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_789_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_790_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_791_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(B,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% add_less_same_cancel1
tff(fact_792_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_793_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_794_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_795_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_796_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% add_less_same_cancel2
tff(fact_797_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel1
tff(fact_798_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel1
tff(fact_799_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel1
tff(fact_800_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(plus_plus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel1
tff(fact_801_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ).

% less_add_same_cancel1
tff(fact_802_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel2
tff(fact_803_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(B,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel2
tff(fact_804_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel2
tff(fact_805_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(plus_plus_int(B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel2
tff(fact_806_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(B,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)) ) ).

% less_add_same_cancel2
tff(fact_807_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_808_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_809_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_810_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_811_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_812_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_813_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_814_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_815_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y)) = zero_z3403309356797280102nteger )
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_816_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_817_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_818_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_819_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_820_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_821_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_822_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_823_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_824_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_825_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_826_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),numeral_numeral_rat(V)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_827_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_828_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_829_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_830_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_831_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_832_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self1
tff(fact_833_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_834_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_835_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self2
tff(fact_836_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_837_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_838_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self3
tff(fact_839_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_840_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_841_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self4
tff(fact_842_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_843_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_844_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_845_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_846_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,bit0(one))) = M ).

% add_self_div_2
tff(fact_847_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_848_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_849_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_z3403309356797280102nteger )
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_850_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_851_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_852_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_853_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_854_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)
tff(fact_855_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% is_num_normalize(1)
tff(fact_856_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_857_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
tff(fact_858_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_859_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_860_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_861_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
tff(fact_862_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_863_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_864_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_865_ab__semigroup__add__class_Oadd_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)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_866_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_867_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(B),aa_int_int(plus_plus_int(A),C)) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_868_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_869_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,B) = plus_plus_rat(B,A) ).

% ab_semigroup_add_class.add.commute
tff(fact_870_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_871_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_872_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_873_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
tff(fact_874_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_875_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_876_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
tff(fact_877_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_878_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_879_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
tff(fact_880_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_881_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% add.assoc
tff(fact_882_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_883_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = plus_plus_rat(K,B) )
     => ( plus_plus_rat(A,B2) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).

% group_cancel.add2
tff(fact_884_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_885_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(plus_plus_int(A),B2) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add2
tff(fact_886_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_887_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( plus_plus_rat(A2,B) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).

% group_cancel.add1
tff(fact_888_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_889_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add1
tff(fact_890_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_891_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( plus_plus_rat(I,K) = plus_plus_rat(J2,L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_892_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_893_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_int_int(plus_plus_int(I),K) = aa_int_int(plus_plus_int(J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_894_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_895_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_896_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% comm_monoid_add_class.add_0
tff(fact_897_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_898_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_899_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,A) = A ).

% comm_monoid_add_class.add_0
tff(fact_900_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_901_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_902_add_Ocomm__neutral,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_903_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_904_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_905_add_Ocomm__neutral,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,zero_z3403309356797280102nteger) = A ).

% add.comm_neutral
tff(fact_906_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_907_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_908_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% add.group_left_neutral
tff(fact_909_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_910_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,A) = A ).

% add.group_left_neutral
tff(fact_911_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_912_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_913_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_914_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_915_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J2))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J2,L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_916_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_917_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_918_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_919_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_920_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J2 )
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J2,L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_921_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J2))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_922_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J2))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_923_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_924_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J2))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_925_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J2))
        & ( K = L ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J2,L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_926_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_strict_mono
tff(fact_927_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_strict_mono
tff(fact_928_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_929_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_strict_mono
tff(fact_930_add__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,D))) ) ) ).

% add_strict_mono
tff(fact_931_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_strict_left_mono
tff(fact_932_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))) ) ).

% add_strict_left_mono
tff(fact_933_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_strict_left_mono
tff(fact_934_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))) ) ).

% add_strict_left_mono
tff(fact_935_add__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(C,A)),plus_p5714425477246183910nteger(C,B))) ) ).

% add_strict_left_mono
tff(fact_936_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_strict_right_mono
tff(fact_937_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))) ) ).

% add_strict_right_mono
tff(fact_938_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_strict_right_mono
tff(fact_939_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))) ) ).

% add_strict_right_mono
tff(fact_940_add__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,C))) ) ).

% add_strict_right_mono
tff(fact_941_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_942_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_943_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_944_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_945_add__less__imp__less__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(C,A)),plus_p5714425477246183910nteger(C,B)))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% add_less_imp_less_left
tff(fact_946_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_947_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_948_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_949_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_950_add__less__imp__less__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,C)))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% add_less_imp_less_right
tff(fact_951_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_952_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_953_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_954_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_955_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_956_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_957_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_958_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_959_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_960_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% comm_semiring_class.distrib
tff(fact_961_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_962_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_963_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_964_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% distrib_left
tff(fact_965_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_966_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_967_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_968_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% distrib_right
tff(fact_969_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E)),C) ).

% combine_common_factor
tff(fact_970_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : plus_plus_rat(aa_rat_rat(times_times_rat(A),E),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),C)) = plus_plus_rat(aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),E),C) ).

% combine_common_factor
tff(fact_971_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_972_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),E)),C) ).

% combine_common_factor
tff(fact_973_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_974_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_975_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
tff(fact_976_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_977_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_978_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_979_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_980_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_981_add__lessD1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_982_add__less__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ) ).

% add_less_mono
tff(fact_983_not__add__less1,axiom,
    ! [I: nat,J2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),I)) ).

% not_add_less1
tff(fact_984_not__add__less2,axiom,
    ! [J2: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I)),I)) ).

% not_add_less2
tff(fact_985_add__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K))) ) ).

% add_less_mono1
tff(fact_986_trans__less__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M))) ) ).

% trans_less_add1
tff(fact_987_trans__less__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J2))) ) ).

% trans_less_add2
tff(fact_988_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_989_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% add_mult_distrib
tff(fact_990_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% add_mult_distrib2
tff(fact_991_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),U)),K) ).

% left_add_mult_distrib
tff(fact_992_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_993_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_994_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_995_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_996_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger)) ) ) ).

% add_neg_neg
tff(fact_997_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_pos
tff(fact_998_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).

% add_pos_pos
tff(fact_999_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_pos
tff(fact_1000_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B))) ) ) ).

% add_pos_pos
tff(fact_1001_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B))) ) ) ).

% add_pos_pos
tff(fact_1002_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1003_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% pos_add_strict
tff(fact_1004_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),plus_plus_rat(A,C))) ) ) ).

% pos_add_strict
tff(fact_1005_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% pos_add_strict
tff(fact_1006_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(plus_plus_int(A),C))) ) ) ).

% pos_add_strict
tff(fact_1007_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),plus_p5714425477246183910nteger(A,C))) ) ) ).

% pos_add_strict
tff(fact_1008_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_1009_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(X,Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_1010_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_1011_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(X,Y)),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),zero_z3403309356797280102nteger)) ) ) ).

% add_less_zeroD
tff(fact_1012_numeral__Bit0,axiom,
    ! [N: num] : numera6690914467698888265omplex(bit0(N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_Bit0
tff(fact_1013_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_real(bit0(N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_Bit0
tff(fact_1014_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_rat(bit0(N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).

% numeral_Bit0
tff(fact_1015_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,bit0(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_1016_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,bit0(N)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_1017_power__add,axiom,
    ! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).

% power_add
tff(fact_1018_power__add,axiom,
    ! [A: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_add
tff(fact_1019_power__add,axiom,
    ! [A: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_add
tff(fact_1020_power__add,axiom,
    ! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).

% power_add
tff(fact_1021_power__add,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_1022_power__add,axiom,
    ! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_1023_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1024_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1025_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_1026_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_1027_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_1028_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1029_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1030_less__imp__add__positive,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_1031_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_1032_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_1033_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_1034_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_1035_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_1036_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_1037_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y))),zero_z3403309356797280102nteger)) ).

% not_sum_squares_lt_zero
tff(fact_1038_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1039_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1040_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1041_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y))))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1042_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1043_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1044_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(A,Z),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(A,Z),B) = divide_divide_rat(plus_plus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1045_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1046_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1047_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( plus_plus_rat(A,divide_divide_rat(B,Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(A,divide_divide_rat(B,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1048_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_1049_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_1050_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_1051_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_1052_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_1053_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(X,Y),Z) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_1054_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_1055_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_1056_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(Z,divide_divide_rat(X,Y)) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_1057_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_1058_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_1059_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_1060_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_1061_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_1062_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_1063_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,bit0(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_1064_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_1065_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1066_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1067_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1068_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1069_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
    <=> ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
tff(fact_1070_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_1071_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_1072_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_1073_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% mult.assoc
tff(fact_1074_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1075_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1076_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1077_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1078_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1079_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1080_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1081_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(times_times_int(B),aa_int_int(times_times_int(A),C)) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1082_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_real] : size_size_list_real(Xs2) = N ).

% Ex_list_of_length
tff(fact_1083_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).

% Ex_list_of_length
tff(fact_1084_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_1085_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_1086_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) != size_size_list_real(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1087_neq__if__length__neq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) != size_size_list_bool(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1088_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1089_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1090_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_1091_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_1092_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),Z) = plus_plus_rat(Z,Z) ).

% mult_2
tff(fact_1093_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_1094_mult__2,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Z) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2
tff(fact_1095_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(bit0(one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_1096_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_1097_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(bit0(one))) = plus_plus_rat(Z,Z) ).

% mult_2_right
tff(fact_1098_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_1099_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2_right
tff(fact_1100_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_1101_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_1102_left__add__twice,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(A,B)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),A),B) ).

% left_add_twice
tff(fact_1103_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),B) ).

% left_add_twice
tff(fact_1104_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),B) ).

% left_add_twice
tff(fact_1105_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I2)),J3) )
               => pp(aa_nat_bool(P,I2)) ) ) ) ) ) ).

% split_div
tff(fact_1106_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1107_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).

% dividend_less_times_div
tff(fact_1108_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),K) ) ) ).

% msrevs(1)
tff(fact_1109_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1110_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1111_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1112_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1113_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1114_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1115_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N: nat] : divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1116_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1117_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_1118_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_1119_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_1120_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_1121_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_z3403309356797280102nteger)) ).

% not_sum_power2_lt_zero
tff(fact_1122_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1123_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1124_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1125_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1126_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(X,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_1127_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_1128_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_1129_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(plus_plus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = plus_plus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_1130_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_1131_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(plus_plus_int(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_1132_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),Y)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Sz))) ) ) ) ).

% nat_add_offset_less
tff(fact_1133_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_1134_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_1135_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1136_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1137_forall__finite_I1_J,axiom,
    ! [P: fun_nat_bool,I4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),zero_zero_nat))
     => pp(aa_nat_bool(P,I4)) ) ).

% forall_finite(1)
tff(fact_1138_length__induct,axiom,
    ! [P: fun_list_real_bool,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2)))
             => pp(aa_list_real_bool(P,Ys2)) )
         => pp(aa_list_real_bool(P,Xs2)) )
     => pp(aa_list_real_bool(P,Xs)) ) ).

% length_induct
tff(fact_1139_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys2: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys2)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P,Ys2)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_1140_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_1141_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_1142_forall__finite_I3_J,axiom,
    ! [X: nat,P: fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(suc,X))))
         => pp(aa_nat_bool(P,I2)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,X)))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I2))) ) ) ) ).

% forall_finite(3)
tff(fact_1143_forall__finite_I2_J,axiom,
    ! [P: fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(suc,zero_zero_nat)))
         => pp(aa_nat_bool(P,I2)) )
    <=> pp(aa_nat_bool(P,zero_zero_nat)) ) ).

% forall_finite(2)
tff(fact_1144_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1145_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,bit0(V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1146_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(bit0(one))))) ) ).

% field_less_half_sum
tff(fact_1147_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),divide_divide_rat(plus_plus_rat(X,Y),numeral_numeral_rat(bit0(one))))) ) ).

% field_less_half_sum
tff(fact_1148_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(bit0(one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(bit0(one)))) = X ).

% field_sum_of_halves
tff(fact_1149_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
tff(fact_1150_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1151_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat(A,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1152_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(plus_plus_int(A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1153_double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% double_eq_0_iff
tff(fact_1154_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_1155_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_1156_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(X)),some_nat(Y))) ) ).

% greater_shift
tff(fact_1157_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(X)),some_nat(Y))) ) ).

% less_shift
tff(fact_1158_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),X)),N) = X ) ) ).

% low_inv
tff(fact_1159_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_1160_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_1161_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_1162_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_1163_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bit0(M)),bit0(N)) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(6)
tff(fact_1164_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = bit0(one) ).

% semiring_norm(2)
tff(fact_1165_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% Suc_numeral
tff(fact_1166_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_1167_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N: num,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_1168_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_1169_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_1170_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_1171_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).

% power_add_numeral2
tff(fact_1172_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_1173_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_1174_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_1175_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_1176_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_1177_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_1178_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(plus_plus_int(K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_1179_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(plus_plus_int(zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_1180_add__One__commute,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N) = aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one) ).

% add_One_commute
tff(fact_1181_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_1182_zdiv__mult__self,axiom,
    ! [M: int,A: int,N: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(M),N))),M) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),M)),N) ) ) ).

% zdiv_mult_self
tff(fact_1183_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(times_times_int(zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_1184_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(times_times_int(K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_1185_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_1186_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1187_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1188_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D2))
       => ? [E2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1189_zmult__zless__mono2,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(K),I)),aa_int_int(times_times_int(K),J2))) ) ) ).

% zmult_zless_mono2
tff(fact_1190_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_1191_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1192_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1193_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_1194_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,product_Pair_int_int(Q2,R2)))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_1195_divides__aux__eq,axiom,
    ! [Q2: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q2,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_1196_low__def,axiom,
    ! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% low_def
tff(fact_1197_mlex__snd__decrI,axiom,
    ! [A: nat,A3: nat,B: nat,B3: nat,N4: nat] :
      ( ( A = A3 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N4)),B3))) ) ) ).

% mlex_snd_decrI
tff(fact_1198_mlex__fst__decrI,axiom,
    ! [A: nat,A3: nat,B: nat,N4: nat,B3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N4))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B3),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N4)),B3))) ) ) ) ).

% mlex_fst_decrI
tff(fact_1199_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N4))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N4))) ) ) ).

% mlex_bound
tff(fact_1200_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1201_option_Osize__gen_I2_J,axiom,
    ! [X: fun_nat_nat,X22: nat] : size_option_nat(X,some_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1202_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X22: num] : size_option_num(X,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1203_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_1204_uint32_Osize__eq,axiom,
    ! [P3: uint32] : size_size_uint32(P3) = aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(one)))))) ).

% uint32.size_eq
tff(fact_1205_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1206_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(R2),C)) != plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1207_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1208_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,aa_rat_rat(times_times_rat(R2),C)) != plus_plus_rat(B,aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1209_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1210_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(R2),C)) != aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1211_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_1212_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_1213_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(X),Z)),aa_int_int(times_times_int(Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_1214_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),Z)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Z)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_1215_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,bit0(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_1216_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_1217_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mod_trivial
tff(fact_1218_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_1219_dvd__0__right,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),zero_zero_complex)) ).

% dvd_0_right
tff(fact_1220_dvd__0__right,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).

% dvd_0_right
tff(fact_1221_dvd__0__right,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),zero_zero_rat)) ).

% dvd_0_right
tff(fact_1222_dvd__0__right,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% dvd_0_right
tff(fact_1223_dvd__0__right,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).

% dvd_0_right
tff(fact_1224_dvd__0__right,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger)) ).

% dvd_0_right
tff(fact_1225_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_1226_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_1227_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_1228_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_1229_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_1230_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_1231_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_1232_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1233_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1234_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_1235_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1236_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1237_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_1238_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_1239_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1240_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_1241_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_1242_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_1243_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_1244_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_1245_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_1246_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_1247_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_1248_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_1249_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_1250_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_1251_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_1252_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_1253_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_1254_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_1255_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_1256_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_1257_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_1258_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_1259_nat__mod__eq_H,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = A ) ) ).

% nat_mod_eq'
tff(fact_1260_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_1261_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_1262_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_1263_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_1264_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_1265_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_1266_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)))
    <=> ( ( C = zero_zero_complex )
        | pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_1267_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_1268_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_1269_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_1270_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_1271_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1272_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1273_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1274_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1275_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1276_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1277_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_1278_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_1279_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_1280_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_1281_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_1282_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_1283_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1284_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(aa_rat_rat(times_times_rat(C),A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1285_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1286_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),A)),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1287_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1288_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,aa_rat_rat(times_times_rat(C),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1289_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1290_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(C),A))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1291_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_1292_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_1293_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_1294_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_1295_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_1296_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_1297_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_1298_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_1299_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_1300_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_1301_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_1302_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_1303_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_1304_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_1305_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_1306_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_1307_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_1308_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_1309_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_1310_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_1311_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_1312_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_1313_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_1314_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_1315_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_1316_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_1317_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_1318_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_1319_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_1320_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_1321_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1322_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_1323_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_1324_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_1325_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_1326_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) ) ) ).

% odd_add
tff(fact_1327_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B)))
    <=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B)) ) ) ).

% odd_add
tff(fact_1328_even__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) ) ) ).

% even_add
tff(fact_1329_even__add,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B)) ) ) ).

% even_add
tff(fact_1330_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) ) ) ).

% even_mult_iff
tff(fact_1331_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(times_times_int(A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B)) ) ) ).

% even_mult_iff
tff(fact_1332_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) ) ).

% even_mod_2_iff
tff(fact_1333_even__mod__2__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)) ) ).

% even_mod_2_iff
tff(fact_1334_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)) ) ).

% even_mod_2_iff
tff(fact_1335_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ).

% even_Suc
tff(fact_1336_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_1337_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% mod2_Suc_Suc
tff(fact_1338_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1339_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% odd_Suc_div_two
tff(fact_1340_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_Suc_div_two
tff(fact_1341_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1342_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_1343_even__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_1344_even__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% even_power
tff(fact_1345_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_1346_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_1347_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_1348_power__less__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),zero_z3403309356797280102nteger))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ) ).

% power_less_zero_eq
tff(fact_1349_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1350_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1351_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1352_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1353_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1354_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1355_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1356_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1357_int__distrib_I1_J,axiom,
    ! [Z1: int,Z2: int,W: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(Z1),Z2)),W) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z2),W)) ).

% int_distrib(1)
tff(fact_1358_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z2: int] : aa_int_int(times_times_int(W),aa_int_int(plus_plus_int(Z1),Z2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z2)) ).

% int_distrib(2)
tff(fact_1359_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_1360_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_1361_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_1362_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1363_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1364_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1365_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_1366_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_1367_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_1368_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) ) ).

% dvd_mod
tff(fact_1369_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(modulo_modulo_int(M),N))) ) ) ).

% dvd_mod
tff(fact_1370_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(M,N))) ) ) ).

% dvd_mod
tff(fact_1371_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_1372_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),C) = aa_int_int(modulo_modulo_int(A),C) ) ) ).

% mod_mod_cancel
tff(fact_1373_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_1374_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_1375_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_1376_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_trans
tff(fact_1377_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_trans
tff(fact_1378_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_1379_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_1380_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_1381_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1382_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1383_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1384_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_1385_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1386_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_1387_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_eq
tff(fact_1388_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
tff(fact_1389_mod__add__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_1390_mod__add__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_1391_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B3),C) ) ) ) ).

% mod_add_cong
tff(fact_1392_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_1393_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_left_eq
tff(fact_1394_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
tff(fact_1395_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_1396_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_right_eq
tff(fact_1397_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
tff(fact_1398_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_1399_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_eq
tff(fact_1400_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ).

% mod_mult_eq
tff(fact_1401_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1402_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1403_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A3),B3),C) ) ) ) ).

% mod_mult_cong
tff(fact_1404_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_1405_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),B)),C) ).

% mod_mult_mult2
tff(fact_1406_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_1407_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_1408_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(times_times_int(C),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ).

% mult_mod_right
tff(fact_1409_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ).

% mult_mod_right
tff(fact_1410_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_1411_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_left_eq
tff(fact_1412_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ).

% mod_mult_left_eq
tff(fact_1413_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_1414_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_right_eq
tff(fact_1415_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ).

% mod_mult_right_eq
tff(fact_1416_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),N)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B) ).

% power_mod
tff(fact_1417_power__mod,axiom,
    ! [A: int,B: int,N: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),N)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),N)),B) ).

% power_mod
tff(fact_1418_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),N),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N),B) ).

% power_mod
tff(fact_1419_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_1420_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_1421_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_1422_dvd__0__left,axiom,
    ! [A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_1423_dvd__0__left,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_1424_dvd__0__left,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_1425_dvd__0__left,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_1426_dvd__0__left,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_1427_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A))
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_1428_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_1429_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_1430_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ) ).

% dvd_add
tff(fact_1431_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C))) ) ) ).

% dvd_add
tff(fact_1432_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ) ).

% dvd_add
tff(fact_1433_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C))) ) ) ).

% dvd_add
tff(fact_1434_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_1435_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_1436_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_1437_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_1438_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_1439_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_1440_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_1441_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_1442_dvdE,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_1443_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ~ ! [K2: rat] : A != aa_rat_rat(times_times_rat(B),K2) ) ).

% dvdE
tff(fact_1444_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_1445_dvdE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ~ ! [K2: int] : A != aa_int_int(times_times_int(B),K2) ) ).

% dvdE
tff(fact_1446_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_1447_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_1448_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_1449_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(times_times_int(B),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_1450_dvd__def,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_1451_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_1452_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_1453_dvd__def,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
    <=> ? [K3: int] : A = aa_int_int(times_times_int(B),K3) ) ).

% dvd_def
tff(fact_1454_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult
tff(fact_1455_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult
tff(fact_1456_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult
tff(fact_1457_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))) ) ).

% dvd_mult
tff(fact_1458_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult2
tff(fact_1459_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult2
tff(fact_1460_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult2
tff(fact_1461_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))) ) ).

% dvd_mult2
tff(fact_1462_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).

% dvd_mult_left
tff(fact_1463_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ).

% dvd_mult_left
tff(fact_1464_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).

% dvd_mult_left
tff(fact_1465_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).

% dvd_mult_left
tff(fact_1466_dvd__triv__left,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ).

% dvd_triv_left
tff(fact_1467_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B))) ).

% dvd_triv_left
tff(fact_1468_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ).

% dvd_triv_left
tff(fact_1469_dvd__triv__left,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(A),B))) ).

% dvd_triv_left
tff(fact_1470_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_1471_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_1472_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_1473_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_1474_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C)) ) ).

% dvd_mult_right
tff(fact_1475_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C)) ) ).

% dvd_mult_right
tff(fact_1476_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ).

% dvd_mult_right
tff(fact_1477_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ).

% dvd_mult_right
tff(fact_1478_dvd__triv__right,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))) ).

% dvd_triv_right
tff(fact_1479_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A))) ).

% dvd_triv_right
tff(fact_1480_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ).

% dvd_triv_right
tff(fact_1481_dvd__triv__right,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),A))) ).

% dvd_triv_right
tff(fact_1482_nat__mod__eq,axiom,
    ! [B: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = B ) ) ) ).

% nat_mod_eq
tff(fact_1483_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1484_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1485_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1486_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1487_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1488_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
       => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1489_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1490_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1491_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1492_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1493_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_1494_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_1495_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N))) ) ).

% dvd_power_same
tff(fact_1496_dvd__power__same,axiom,
    ! [X: real,Y: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) ) ).

% dvd_power_same
tff(fact_1497_dvd__power__same,axiom,
    ! [X: int,Y: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N))) ) ).

% dvd_power_same
tff(fact_1498_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N))) ) ).

% dvd_power_same
tff(fact_1499_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N))) ) ).

% dvd_power_same
tff(fact_1500_dvd__power__same,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N))) ) ).

% dvd_power_same
tff(fact_1501_mod__plus__right,axiom,
    ! [A: nat,X: nat,M: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),X)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),X)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),M) ) ) ).

% mod_plus_right
tff(fact_1502_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1503_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1504_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1505_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1506_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1507_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1508_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(M))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1509_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(N))),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1510_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(M)),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(N)),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1511_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1512_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1513_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_1514_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1515_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1516_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
tff(fact_1517_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D3: int] : B != aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),D3)) ) ).

% mod_eqE
tff(fact_1518_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D3: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),D3)) ) ).

% mod_eqE
tff(fact_1519_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_1520_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_1521_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
tff(fact_1522_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1523_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1524_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1525_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1526_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1527_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1528_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).

% mod_Suc
tff(fact_1529_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_1530_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1531_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1532_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1533_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1534_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_1535_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_1536_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_1537_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),C) ) ) ).

% div_mult_swap
tff(fact_1538_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_1539_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_1540_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_1541_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(B),C)),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_1542_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))) ) ).

% dvd_mult_imp_div
tff(fact_1543_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))) ) ).

% dvd_mult_imp_div
tff(fact_1544_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),C))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1545_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),C))
       => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1546_nat__mod__lem,axiom,
    ! [N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),N))
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) = B ) ) ) ).

% nat_mod_lem
tff(fact_1547_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_less_divisor
tff(fact_1548_div__power,axiom,
    ! [B: code_integer,A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),N) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ) ) ).

% div_power
tff(fact_1549_div__power,axiom,
    ! [B: nat,A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ) ) ).

% div_power
tff(fact_1550_div__power,axiom,
    ! [B: int,A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ) ) ).

% div_power
tff(fact_1551_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),L))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),N)),L) = N ) ) ) ).

% word_rot_lem
tff(fact_1552_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_1553_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) ).

% msrevs(2)
tff(fact_1554_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1555_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_1556_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1557_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(N))),aa_num_int(numeral_numeral_int,bit0(Q2))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1558_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(N)),numera6620942414471956472nteger(bit0(Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1559_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_1560_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_1561_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_1562_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(M))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1563_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1564_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(M)),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1565_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1566_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(N))),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1567_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(N)),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1568_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1569_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1570_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_1571_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1572_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1573_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_1574_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_1575_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_1576_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_1577_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_1578_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_1579_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_1580_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1581_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1582_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1583_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_1584_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_1585_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_1586_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_1587_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_1588_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_1589_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_1590_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_1591_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_1592_even__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,bit0(N)))) ).

% even_numeral
tff(fact_1593_even__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_num_int(numeral_numeral_int,bit0(N)))) ).

% even_numeral
tff(fact_1594_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1595_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1596_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(times_times_int(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1597_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1598_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1599_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1600_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1601_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1602_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1603_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),D))
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1604_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1605_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(times_times_int(B),C) = aa_int_int(times_times_int(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1606_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_1607_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_1608_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1609_bex2I,axiom,
    ! [A: heap_e7401611519738050253t_unit,B: set_nat,S2: set_Pr3948176798113811640et_nat,P: fun_he7710493197641524256t_bool] :
      ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A,B),S2)
     => ( ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A,B),S2)
         => pp(aa_set_nat_bool(aa_hea2782697410307197849t_bool(P,A),B)) )
       => ? [A4: heap_e7401611519738050253t_unit,B4: set_nat] :
            ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A4,B4),S2)
            & pp(aa_set_nat_bool(aa_hea2782697410307197849t_bool(P,A4),B4)) ) ) ) ).

% bex2I
tff(fact_1610_bex2I,axiom,
    ! [A: num,B: num,S2: set_Pr8218934625190621173um_num,P: fun_num_fun_num_bool] :
      ( member7279096912039735102um_num(product_Pair_num_num(A,B),S2)
     => ( ( member7279096912039735102um_num(product_Pair_num_num(A,B),S2)
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) )
       => ? [A4: num,B4: num] :
            ( member7279096912039735102um_num(product_Pair_num_num(A4,B4),S2)
            & pp(aa_num_bool(aa_num_fun_num_bool(P,A4),B4)) ) ) ) ).

% bex2I
tff(fact_1611_bex2I,axiom,
    ! [A: nat,B: num,S2: set_Pr6200539531224447659at_num,P: fun_nat_fun_num_bool] :
      ( member9148766508732265716at_num(product_Pair_nat_num(A,B),S2)
     => ( ( member9148766508732265716at_num(product_Pair_nat_num(A,B),S2)
         => pp(aa_num_bool(aa_nat_fun_num_bool(P,A),B)) )
       => ? [A4: nat,B4: num] :
            ( member9148766508732265716at_num(product_Pair_nat_num(A4,B4),S2)
            & pp(aa_num_bool(aa_nat_fun_num_bool(P,A4),B4)) ) ) ) ).

% bex2I
tff(fact_1612_bex2I,axiom,
    ! [A: nat,B: nat,S2: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_bool] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),S2)
     => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),S2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) )
       => ? [A4: nat,B4: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4),S2)
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4)) ) ) ) ).

% bex2I
tff(fact_1613_bex2I,axiom,
    ! [A: int,B: int,S2: set_Pr958786334691620121nt_int,P: fun_int_fun_int_bool] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),S2)
     => ( ( member5262025264175285858nt_int(product_Pair_int_int(A,B),S2)
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) )
       => ? [A4: int,B4: int] :
            ( member5262025264175285858nt_int(product_Pair_int_int(A4,B4),S2)
            & pp(aa_int_bool(aa_int_fun_int_bool(P,A4),B4)) ) ) ) ).

% bex2I
tff(fact_1614_even__zero,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),zero_z3403309356797280102nteger)) ).

% even_zero
tff(fact_1615_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),zero_zero_nat)) ).

% even_zero
tff(fact_1616_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),zero_zero_int)) ).

% even_zero
tff(fact_1617_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% odd_even_add
tff(fact_1618_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B))) ) ) ).

% odd_even_add
tff(fact_1619_evenE,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B4) ) ).

% evenE
tff(fact_1620_evenE,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ~ ! [B4: int] : A != aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B4) ) ).

% evenE
tff(fact_1621_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ) ).

% bit_eq_rec
tff(fact_1622_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B)) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ) ).

% bit_eq_rec
tff(fact_1623_mod__lemma,axiom,
    ! [C: nat,R2: nat,B: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q2),C))),R2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mod_lemma
tff(fact_1624_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I2)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1625_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_1626_even__two__times__div__two,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_1627_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_1628_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se4203085406695923979it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1629_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),bit_se8260200283734997820nteger(M,A)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1630_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1631_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_1632_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1633_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),B))
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1634_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))),B))
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1635_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_1636_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))) = A )
     => ( aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_1637_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
tff(fact_1638_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_1639_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_1640_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_1641_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1642_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),B))
       => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1643_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))),B))
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1644_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_1645_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1646_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = plus_plus_rat(B,A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_1647_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_1648_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(plus_plus_int(B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_1649_add__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B = plus_p5714425477246183910nteger(B,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% add_0_iff
tff(fact_1650_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1651_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(W),Y),aa_rat_rat(times_times_rat(X),Z)) = plus_plus_rat(aa_rat_rat(times_times_rat(W),Z),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1652_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1653_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Y)),aa_int_int(times_times_int(X),Z)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z)),aa_int_int(times_times_int(X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1654_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1655_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),D)) != plus_plus_rat(aa_rat_rat(times_times_rat(A),D),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1656_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1657_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) != aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1658_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_1659_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_1660_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_1661_zero__less__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
    <=> ( ( N = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_1662_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% unset_bit_Suc
tff(fact_1663_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_1664_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% unset_bit_Suc
tff(fact_1665_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_1666_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_1667_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_1668_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))))) ) ).

% even_even_mod_4_iff
tff(fact_1669_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1670_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).

% div_mod_decomp
tff(fact_1671_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ) ) ).

% div_less_mono
tff(fact_1672_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X2: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),B))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1673_unity__coeff__ex,axiom,
    ! [P: fun_complex_bool,L: complex] :
      ( ? [X5: complex] : pp(aa_complex_bool(P,aa_complex_complex(times_times_complex(L),X5)))
    <=> ? [X5: complex] :
          ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X5),zero_zero_complex)))
          & pp(aa_complex_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_1674_unity__coeff__ex,axiom,
    ! [P: fun_Co8822979462182099027r_bool,L: code_integer] :
      ( ? [X5: code_integer] : pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(L),X5)))
    <=> ? [X5: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X5,zero_z3403309356797280102nteger)))
          & pp(aa_Code_integer_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_1675_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X5: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X5)))
    <=> ? [X5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),zero_zero_real)))
          & pp(aa_real_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_1676_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X5: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X5)))
    <=> ? [X5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),plus_plus_rat(X5,zero_zero_rat)))
          & pp(aa_rat_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_1677_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X5: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X5)))
    <=> ? [X5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),zero_zero_nat)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_1678_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X5: int] : pp(aa_int_bool(P,aa_int_int(times_times_int(L),X5)))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(plus_plus_int(X5),zero_zero_int)))
          & pp(aa_int_bool(P,X5)) ) ) ).

% unity_coeff_ex
tff(fact_1679_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_1680_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% flip_bit_Suc
tff(fact_1681_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% flip_bit_Suc
tff(fact_1682_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% set_bit_Suc
tff(fact_1683_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% set_bit_Suc
tff(fact_1684_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% set_bit_Suc
tff(fact_1685_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( bit0(X22) = bit0(Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1686_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_1687_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_1688_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_1689_mod__plus__cong,axiom,
    ! [B: int,B3: int,X: int,X6: int,Y: int,Y5: int,Z3: int] :
      ( ( B = B3 )
     => ( ( aa_int_int(modulo_modulo_int(X),B3) = aa_int_int(modulo_modulo_int(X6),B3) )
       => ( ( aa_int_int(modulo_modulo_int(Y),B3) = aa_int_int(modulo_modulo_int(Y5),B3) )
         => ( ( aa_int_int(plus_plus_int(X6),Y5) = Z3 )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(X),Y)),B) = aa_int_int(modulo_modulo_int(Z3),B3) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_1690_zmod__helper,axiom,
    ! [N: int,M: int,K: int,A: int] :
      ( ( aa_int_int(modulo_modulo_int(N),M) = K )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(N),A)),M) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(K),A)),M) ) ) ).

% zmod_helper
tff(fact_1691_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),T))) ) ) ).

% zdvd_mono
tff(fact_1692_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1693_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L))) ) ).

% neg_mod_bound
tff(fact_1694_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
     => ? [Q3: int] : M = aa_int_int(times_times_int(D),Q3) ) ).

% zmod_eq_0D
tff(fact_1695_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
    <=> ? [Q4: int] : M = aa_int_int(times_times_int(D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_1696_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => ( ( aa_int_int(modulo_modulo_int(A2),N) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),N) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1697_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_1698_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),N)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_1699_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] : A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),N)),aa_int_int(modulo_modulo_int(A2),N)) ).

% div_mod_decomp_int
tff(fact_1700_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X),aa_int_int(times_times_int(C),D))),T))) ) ) ).

% zdvd_period
tff(fact_1701_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(plus_plus_int(N),aa_int_int(times_times_int(K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N)) ) ).

% zdvd_reduce
tff(fact_1702_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1703_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1704_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1705_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1706_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1707_verit__comp__simplify1_I1_J,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1708_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q5: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X2))
         => ( pp(aa_real_bool(P,X2))
          <=> pp(aa_real_bool(P2,X2)) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X2))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q5,X2)) ) )
       => ? [Z5: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
           => ( ( pp(aa_real_bool(P,X3))
                & pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P2,X3))
                & pp(aa_real_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1709_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q5: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X2))
         => ( pp(aa_rat_bool(P,X2))
          <=> pp(aa_rat_bool(P2,X2)) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X2))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q5,X2)) ) )
       => ? [Z5: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
           => ( ( pp(aa_rat_bool(P,X3))
                & pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P2,X3))
                & pp(aa_rat_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1710_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q5: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q5,X2)) ) )
       => ? [Z5: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
           => ( ( pp(aa_num_bool(P,X3))
                & pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P2,X3))
                & pp(aa_num_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1711_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q5: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X2))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q5,X2)) ) )
       => ? [Z5: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
           => ( ( pp(aa_nat_bool(P,X3))
                & pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P2,X3))
                & pp(aa_nat_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1712_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q5: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X2))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q5,X2)) ) )
       => ? [Z5: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
           => ( ( pp(aa_int_bool(P,X3))
                & pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P2,X3))
                & pp(aa_int_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1713_pinf_I1_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q5: fun_Co8822979462182099027r_bool] :
      ( ? [Z4: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z4),X2))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z4: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z4),X2))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q5,X2)) ) )
       => ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
           => ( ( pp(aa_Code_integer_bool(P,X3))
                & pp(aa_Code_integer_bool(Q,X3)) )
            <=> ( pp(aa_Code_integer_bool(P2,X3))
                & pp(aa_Code_integer_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1714_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q5: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X2))
         => ( pp(aa_real_bool(P,X2))
          <=> pp(aa_real_bool(P2,X2)) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z4),X2))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q5,X2)) ) )
       => ? [Z5: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
           => ( ( pp(aa_real_bool(P,X3))
                | pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P2,X3))
                | pp(aa_real_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1715_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q5: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X2))
         => ( pp(aa_rat_bool(P,X2))
          <=> pp(aa_rat_bool(P2,X2)) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z4),X2))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q5,X2)) ) )
       => ? [Z5: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
           => ( ( pp(aa_rat_bool(P,X3))
                | pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P2,X3))
                | pp(aa_rat_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1716_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q5: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z4),X2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q5,X2)) ) )
       => ? [Z5: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
           => ( ( pp(aa_num_bool(P,X3))
                | pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P2,X3))
                | pp(aa_num_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1717_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q5: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X2))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q5,X2)) ) )
       => ? [Z5: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
           => ( ( pp(aa_nat_bool(P,X3))
                | pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P2,X3))
                | pp(aa_nat_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1718_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q5: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X2))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q5,X2)) ) )
       => ? [Z5: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
           => ( ( pp(aa_int_bool(P,X3))
                | pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P2,X3))
                | pp(aa_int_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1719_pinf_I2_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q5: fun_Co8822979462182099027r_bool] :
      ( ? [Z4: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z4),X2))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z4: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z4),X2))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q5,X2)) ) )
       => ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
           => ( ( pp(aa_Code_integer_bool(P,X3))
                | pp(aa_Code_integer_bool(Q,X3)) )
            <=> ( pp(aa_Code_integer_bool(P2,X3))
                | pp(aa_Code_integer_bool(Q5,X3)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1720_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1721_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1722_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1723_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1724_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1725_pinf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1726_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1727_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1728_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1729_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1730_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1731_pinf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1732_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),T)) ) ).

% pinf(5)
tff(fact_1733_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),T)) ) ).

% pinf(5)
tff(fact_1734_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),T)) ) ).

% pinf(5)
tff(fact_1735_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),T)) ) ).

% pinf(5)
tff(fact_1736_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T)) ) ).

% pinf(5)
tff(fact_1737_pinf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),T)) ) ).

% pinf(5)
tff(fact_1738_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X3)) ) ).

% pinf(7)
tff(fact_1739_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X3)) ) ).

% pinf(7)
tff(fact_1740_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X3)) ) ).

% pinf(7)
tff(fact_1741_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X3)) ) ).

% pinf(7)
tff(fact_1742_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3)) ) ).

% pinf(7)
tff(fact_1743_pinf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,T),X3)) ) ).

% pinf(7)
tff(fact_1744_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q5: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z4))
         => ( pp(aa_real_bool(P,X2))
          <=> pp(aa_real_bool(P2,X2)) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z4))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q5,X2)) ) )
       => ? [Z5: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
           => ( ( pp(aa_real_bool(P,X3))
                & pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P2,X3))
                & pp(aa_real_bool(Q5,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1745_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q5: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z4))
         => ( pp(aa_rat_bool(P,X2))
          <=> pp(aa_rat_bool(P2,X2)) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z4))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q5,X2)) ) )
       => ? [Z5: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
           => ( ( pp(aa_rat_bool(P,X3))
                & pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P2,X3))
                & pp(aa_rat_bool(Q5,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1746_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q5: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z4))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z4))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q5,X2)) ) )
       => ? [Z5: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
           => ( ( pp(aa_num_bool(P,X3))
                & pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P2,X3))
                & pp(aa_num_bool(Q5,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1747_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q5: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z4))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z4))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q5,X2)) ) )
       => ? [Z5: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
           => ( ( pp(aa_nat_bool(P,X3))
                & pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P2,X3))
                & pp(aa_nat_bool(Q5,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1748_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q5: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z4))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z4))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q5,X2)) ) )
       => ? [Z5: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
           => ( ( pp(aa_int_bool(P,X3))
                & pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P2,X3))
                & pp(aa_int_bool(Q5,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1749_minf_I1_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q5: fun_Co8822979462182099027r_bool] :
      ( ? [Z4: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z4))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z4: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z4))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q5,X2)) ) )
       => ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
           => ( ( pp(aa_Code_integer_bool(P,X3))
                & pp(aa_Code_integer_bool(Q,X3)) )
            <=> ( pp(aa_Code_integer_bool(P2,X3))
                & pp(aa_Code_integer_bool(Q5,X3)) ) ) ) ) ) ).

% minf(1)
tff(fact_1750_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q5: fun_real_bool] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z4))
         => ( pp(aa_real_bool(P,X2))
          <=> pp(aa_real_bool(P2,X2)) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z4))
           => ( pp(aa_real_bool(Q,X2))
            <=> pp(aa_real_bool(Q5,X2)) ) )
       => ? [Z5: real] :
          ! [X3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
           => ( ( pp(aa_real_bool(P,X3))
                | pp(aa_real_bool(Q,X3)) )
            <=> ( pp(aa_real_bool(P2,X3))
                | pp(aa_real_bool(Q5,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1751_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q5: fun_rat_bool] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z4))
         => ( pp(aa_rat_bool(P,X2))
          <=> pp(aa_rat_bool(P2,X2)) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z4))
           => ( pp(aa_rat_bool(Q,X2))
            <=> pp(aa_rat_bool(Q5,X2)) ) )
       => ? [Z5: rat] :
          ! [X3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
           => ( ( pp(aa_rat_bool(P,X3))
                | pp(aa_rat_bool(Q,X3)) )
            <=> ( pp(aa_rat_bool(P2,X3))
                | pp(aa_rat_bool(Q5,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1752_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q5: fun_num_bool] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z4))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z4))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q5,X2)) ) )
       => ? [Z5: num] :
          ! [X3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
           => ( ( pp(aa_num_bool(P,X3))
                | pp(aa_num_bool(Q,X3)) )
            <=> ( pp(aa_num_bool(P2,X3))
                | pp(aa_num_bool(Q5,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1753_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q5: fun_nat_bool] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z4))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z4))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q5,X2)) ) )
       => ? [Z5: nat] :
          ! [X3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
           => ( ( pp(aa_nat_bool(P,X3))
                | pp(aa_nat_bool(Q,X3)) )
            <=> ( pp(aa_nat_bool(P2,X3))
                | pp(aa_nat_bool(Q5,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1754_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q5: fun_int_bool] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z4))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z4))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q5,X2)) ) )
       => ? [Z5: int] :
          ! [X3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
           => ( ( pp(aa_int_bool(P,X3))
                | pp(aa_int_bool(Q,X3)) )
            <=> ( pp(aa_int_bool(P2,X3))
                | pp(aa_int_bool(Q5,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1755_minf_I2_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q5: fun_Co8822979462182099027r_bool] :
      ( ? [Z4: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z4))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z4: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z4))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q5,X2)) ) )
       => ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
           => ( ( pp(aa_Code_integer_bool(P,X3))
                | pp(aa_Code_integer_bool(Q,X3)) )
            <=> ( pp(aa_Code_integer_bool(P2,X3))
                | pp(aa_Code_integer_bool(Q5,X3)) ) ) ) ) ) ).

% minf(2)
tff(fact_1756_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1757_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1758_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1759_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1760_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1761_minf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1762_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1763_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1764_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1765_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1766_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1767_minf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1768_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),T)) ) ).

% minf(5)
tff(fact_1769_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),T)) ) ).

% minf(5)
tff(fact_1770_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),T)) ) ).

% minf(5)
tff(fact_1771_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),T)) ) ).

% minf(5)
tff(fact_1772_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T)) ) ).

% minf(5)
tff(fact_1773_minf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),T)) ) ).

% minf(5)
tff(fact_1774_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X3)) ) ).

% minf(7)
tff(fact_1775_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X3)) ) ).

% minf(7)
tff(fact_1776_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X3)) ) ).

% minf(7)
tff(fact_1777_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X3)) ) ).

% minf(7)
tff(fact_1778_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3)) ) ).

% minf(7)
tff(fact_1779_minf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,T),X3)) ) ).

% minf(7)
tff(fact_1780_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M3: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M3) ).

% prod_decode_aux.cases
tff(fact_1781_mod__exp__less__eq__exp,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ).

% mod_exp_less_eq_exp
tff(fact_1782_pos__mod__bound2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% pos_mod_bound2
tff(fact_1783_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1784_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1785_verit__sum__simplify,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1786_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_1787_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_1788_verit__sum__simplify,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,zero_z3403309356797280102nteger) = A ).

% verit_sum_simplify
tff(fact_1789_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_1790_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_1791_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_1792_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ? [B5: nat,C3: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B5),C3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B5),B))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C3),C)) ) ) ).

% division_decomp
tff(fact_1793_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)))
     => ? [B5: int,C3: int] :
          ( ( A = aa_int_int(times_times_int(B5),C3) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B5),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C3),C)) ) ) ).

% division_decomp
tff(fact_1794_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ~ ! [X2: nat,Y3: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y3) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X2),A))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_1795_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P3),aa_int_int(times_times_int(A),B)))
     => ~ ! [X2: int,Y3: int] :
            ( ( P3 = aa_int_int(times_times_int(X2),Y3) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X2),A))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_1796_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B4),A4)) )
     => ( ! [A4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),zero_zero_nat))
       => ( ! [A4: nat,B4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_1797_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1798_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1799_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_1800_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_1801_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_1802_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se7879613467334960850it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1803_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1804_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se2159334234014336723it_int(M),A)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1805_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1806_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% pinf(9)
tff(fact_1807_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S))) ) ) ).

% pinf(9)
tff(fact_1808_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% pinf(9)
tff(fact_1809_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S))) ) ) ).

% pinf(9)
tff(fact_1810_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))) ) ) ).

% pinf(9)
tff(fact_1811_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% pinf(10)
tff(fact_1812_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S))) ) ) ).

% pinf(10)
tff(fact_1813_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% pinf(10)
tff(fact_1814_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S))) ) ) ).

% pinf(10)
tff(fact_1815_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))) ) ) ).

% pinf(10)
tff(fact_1816_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% minf(9)
tff(fact_1817_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S))) ) ) ).

% minf(9)
tff(fact_1818_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% minf(9)
tff(fact_1819_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S))) ) ) ).

% minf(9)
tff(fact_1820_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))) ) ) ).

% minf(9)
tff(fact_1821_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))) ) ) ).

% minf(10)
tff(fact_1822_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,S))) ) ) ).

% minf(10)
tff(fact_1823_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))) ) ) ).

% minf(10)
tff(fact_1824_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),S))) ) ) ).

% minf(10)
tff(fact_1825_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S)))
      <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X3,S))) ) ) ).

% minf(10)
tff(fact_1826_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_1827_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
      ( ! [M3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),zero_zero_nat))
     => ( ! [M3: nat,N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),N2)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_1828_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D3) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_1829_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X2: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1830_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% triangle_def
tff(fact_1831_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1832_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1833_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1834_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_1835_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_1836_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_1837_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% set_bit_0
tff(fact_1838_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% set_bit_0
tff(fact_1839_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X)) ) ).

% set_decode_0
tff(fact_1840_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% flip_bit_0
tff(fact_1841_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% flip_bit_0
tff(fact_1842_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% flip_bit_0
tff(fact_1843_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X)))
    <=> pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% set_decode_Suc
tff(fact_1844_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1845_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1846_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1847_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1848_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_1849_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_1850_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_1851_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_1852_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(K)),aa_num_nat(numeral_numeral_nat,L)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_1853_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(numeral_numeral_rat(K)),aa_num_nat(numeral_numeral_nat,L)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_1854_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_1855_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_1856_min__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X5)) ) ) ) ).

% min_in_set_def
tff(fact_1857_max__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),X)) ) ) ) ).

% max_in_set_def
tff(fact_1858_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,bit0(M)),bit0(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(71)
tff(fact_1859_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,one),N)) ).

% semiring_norm(68)
tff(fact_1860_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% Suc_le_mono
tff(fact_1861_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_1862_le0,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% le0
tff(fact_1863_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_1864_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_1865_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_1866_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_1867_power__one__right,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),one_one_nat) = A ).

% power_one_right
tff(fact_1868_power__one__right,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),one_one_nat) = A ).

% power_one_right
tff(fact_1869_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_1870_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1871_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1872_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_1873_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1874_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1875_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1876_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1877_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% numeral_le_iff
tff(fact_1878_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1879_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_1880_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_1881_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_1882_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1883_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_1884_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_1885_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_1886_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,bit0(M)),one)) ).

% semiring_norm(69)
tff(fact_1887_mult_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(A),one_one_assn) = A ).

% mult.right_neutral
tff(fact_1888_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_1889_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1890_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_1891_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.right_neutral
tff(fact_1892_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(one_one_assn),A) = A ).

% mult_1
tff(fact_1893_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_1894_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1895_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_1896_mult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% mult_1
tff(fact_1897_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_1898_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_1899_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_1900_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_1901_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_1902_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_1903_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_1904_power__one,axiom,
    ! [N: nat] : aa_nat_assn(power_power_assn(one_one_assn),N) = one_one_assn ).

% power_one
tff(fact_1905_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_1906_power__one,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),N) = one_one_real ).

% power_one
tff(fact_1907_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_1908_power__one,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(one_one_complex),N) = one_one_complex ).

% power_one
tff(fact_1909_power__one,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ).

% power_one
tff(fact_1910_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_1911_less__eq__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ord_le8614940839814719452n_real(some_real(X),some_real(Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% less_eq_option_Some
tff(fact_1912_less__eq__option__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_le2843612097646854710et_nat(some_set_nat(X),some_set_nat(Y))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% less_eq_option_Some
tff(fact_1913_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,some_num(X)),some_num(Y)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% less_eq_option_Some
tff(fact_1914_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,some_nat(X)),some_nat(Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% less_eq_option_Some
tff(fact_1915_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ord_le1736525451366464988on_int(some_int(X),some_int(Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% less_eq_option_Some
tff(fact_1916_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_1917_signed__take__bit__of__0,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(N,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% signed_take_bit_of_0
tff(fact_1918_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_1919_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_1920_of__bool__eq_I1_J,axiom,
    aa_bool_real(zero_n3304061248610475627l_real,fFalse) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_1921_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_1922_of__bool__eq_I1_J,axiom,
    aa_bool_nat(zero_n2687167440665602831ol_nat,fFalse) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_1923_of__bool__eq_I1_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_1924_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger(fFalse) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_1925_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n1201886186963655149omplex(P) = zero_zero_complex )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_1926_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_real(zero_n3304061248610475627l_real,P) = zero_zero_real )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_1927_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_1928_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = zero_zero_nat )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_1929_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_1930_of__bool__eq__0__iff,axiom,
    ! [P: bool] :
      ( ( zero_n356916108424825756nteger(P) = zero_z3403309356797280102nteger )
    <=> ~ pp(P) ) ).

% of_bool_eq_0_iff
tff(fact_1931_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(zero_n3304061248610475627l_real,P)),aa_bool_real(zero_n3304061248610475627l_real,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1932_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1933_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1934_of__bool__less__eq__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)))
    <=> ( pp(P)
       => pp(Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_1935_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_bool_real(zero_n3304061248610475627l_real,P)),aa_bool_real(zero_n3304061248610475627l_real,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_1936_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_1937_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_1938_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_1939_of__bool__less__iff,axiom,
    ! [P: bool,Q: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)))
    <=> ( ~ pp(P)
        & pp(Q) ) ) ).

% of_bool_less_iff
tff(fact_1940_of__bool__eq_I2_J,axiom,
    aa_bool_real(zero_n3304061248610475627l_real,fTrue) = one_one_real ).

% of_bool_eq(2)
tff(fact_1941_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat(fTrue) = one_one_rat ).

% of_bool_eq(2)
tff(fact_1942_of__bool__eq_I2_J,axiom,
    aa_bool_nat(zero_n2687167440665602831ol_nat,fTrue) = one_one_nat ).

% of_bool_eq(2)
tff(fact_1943_of__bool__eq_I2_J,axiom,
    aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).

% of_bool_eq(2)
tff(fact_1944_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger(fTrue) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_1945_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_real(zero_n3304061248610475627l_real,P) = one_one_real )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_1946_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n2052037380579107095ol_rat(P) = one_one_rat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_1947_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P) = one_one_nat )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_1948_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_1949_of__bool__eq__1__iff,axiom,
    ! [P: bool] :
      ( ( zero_n356916108424825756nteger(P) = one_one_Code_integer )
    <=> pp(P) ) ).

% of_bool_eq_1_iff
tff(fact_1950_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% zle_add1_eq_le
tff(fact_1951_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1952_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1953_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1954_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).

% enat_ord_number(1)
tff(fact_1955_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1956_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,some_nat(X)),some_nat(Y))) ) ).

% lesseq_shift
tff(fact_1957_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(B,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_1958_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(B,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% add_le_same_cancel1
tff(fact_1959_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_1960_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_1961_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_1962_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_1963_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,B)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% add_le_same_cancel2
tff(fact_1964_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_1965_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_1966_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_1967_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel1
tff(fact_1968_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),plus_p5714425477246183910nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) ).

% le_add_same_cancel1
tff(fact_1969_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel1
tff(fact_1970_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel1
tff(fact_1971_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(plus_plus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel1
tff(fact_1972_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(B,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel2
tff(fact_1973_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),plus_p5714425477246183910nteger(B,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) ).

% le_add_same_cancel2
tff(fact_1974_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel2
tff(fact_1975_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel2
tff(fact_1976_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(plus_plus_int(B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel2
tff(fact_1977_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1978_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1979_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1980_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1981_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1982_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1983_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1984_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1985_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1986_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_1987_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1988_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1989_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1990_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1991_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_1992_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1993_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1994_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(times_times_int(C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1995_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1996_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_1997_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1998_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1999_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2000_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2001_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_2002_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2003_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2004_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(times_times_int(A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2005_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex(N) = one_one_complex )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2006_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real(N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2007_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2008_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2009_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_2010_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2011_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = numeral_numeral_real(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2012_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2013_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2014_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_2015_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2016_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2017_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
tff(fact_2018_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(A,A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_2019_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2020_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_2021_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_2022_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_2023_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_2024_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2025_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2026_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
tff(fact_2027_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2028_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_2029_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_2030_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_2031_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_2032_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
tff(fact_2033_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2034_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
tff(fact_2035_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2036_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
tff(fact_2037_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2038_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
tff(fact_2039_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2040_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
tff(fact_2041_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2042_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2043_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2044_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2045_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_2046_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_2047_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2048_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_2049_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_2050_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_2051_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_2052_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_2053_unit__prod,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int)) ) ) ).

% unit_prod
tff(fact_2054_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat)) ) ) ).

% unit_div
tff(fact_2055_unit__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)) ) ) ).

% unit_div
tff(fact_2056_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat)) ) ).

% unit_div_1_unit
tff(fact_2057_unit__div__1__unit,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_2058_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2059_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2060_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).

% one_le_mult_iff
tff(fact_2061_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2062_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_2063_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2064_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_bool_real(zero_n3304061248610475627l_real,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_2065_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_2066_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_bool_nat(zero_n2687167440665602831ol_nat,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_2067_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_2068_zero__less__of__bool__iff,axiom,
    ! [P: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P)))
    <=> pp(P) ) ).

% zero_less_of_bool_iff
tff(fact_2069_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_2070_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_bool_real(zero_n3304061248610475627l_real,P)),one_one_real))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_2071_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_2072_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),one_one_nat))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_2073_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_2074_of__bool__less__one__iff,axiom,
    ! [P: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P)),one_one_Code_integer))
    <=> ~ pp(P) ) ).

% of_bool_less_one_iff
tff(fact_2075_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_2076_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_2077_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2078_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2079_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2080_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2081_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_2082_triangle__Suc,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).

% triangle_Suc
tff(fact_2083_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2084_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2085_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2086_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).

% numeral_le_one_iff
tff(fact_2087_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_2088_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_2089_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2090_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_divide_1_iff
tff(fact_2091_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_2092_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_2093_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2094_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_2095_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2096_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_2097_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2098_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),divide_divide_rat(B,A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_2099_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2100_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),divide_divide_rat(B,A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_2101_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2102_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_divide_1_iff
tff(fact_2103_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2104_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2105_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2106_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2107_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2108_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2109_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2110_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2111_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2112_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2113_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2114_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2115_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2116_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2117_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2118_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% Suc_1
tff(fact_2119_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2120_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2121_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2122_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2123_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2124_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_2125_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_2126_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_2127_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_2128_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2129_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2130_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_2131_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2132_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_2133_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2134_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_2135_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2136_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_2137_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2138_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2139_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2140_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2141_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2142_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(bit0(one)) ).

% one_add_one
tff(fact_2143_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(bit0(one)) ).

% one_add_one
tff(fact_2144_one__add__one,axiom,
    plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% one_add_one
tff(fact_2145_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% one_add_one
tff(fact_2146_one__add__one,axiom,
    aa_int_int(plus_plus_int(one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% one_add_one
tff(fact_2147_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2148_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2149_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2150_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2151_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2152_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
          <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2153_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2154_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2155_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2156_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2157_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_2158_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_2159_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_2160_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_2161_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_2162_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
tff(fact_2163_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,bit0(K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_Suc_bit0
tff(fact_2164_odd__of__bool__self,axiom,
    ! [P3: bool] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_bool_nat(zero_n2687167440665602831ol_nat,P3)))
    <=> pp(P3) ) ).

% odd_of_bool_self
tff(fact_2165_odd__of__bool__self,axiom,
    ! [P3: bool] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> pp(P3) ) ).

% odd_of_bool_self
tff(fact_2166_odd__of__bool__self,axiom,
    ! [P3: bool] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(P3)))
    <=> pp(P3) ) ).

% odd_of_bool_self
tff(fact_2167_numeral__plus__one,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_2168_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_2169_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_rat(numeral_numeral_rat(N),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_2170_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_2171_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_2172_one__plus__numeral,axiom,
    ! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_2173_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_2174_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_2175_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_2176_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_2177_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_2178_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),numera6620942414471956472nteger(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2179_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),numeral_numeral_real(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2180_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),numeral_numeral_rat(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2181_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2182_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).

% one_less_numeral_iff
tff(fact_2183_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_2184_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2185_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2186_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_2187_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2188_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2189_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_2190_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2191_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2192_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_2193_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2194_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2195_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2196_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2197_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2198_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2199_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2200_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
tff(fact_2201_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2202_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2203_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
tff(fact_2204_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)) ) ).

% even_plus_one_iff
tff(fact_2205_even__plus__one__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),one_one_int)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)) ) ).

% even_plus_one_iff
tff(fact_2206_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B)),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_2207_of__bool__half__eq__0,axiom,
    ! [B: bool] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_bool_int(zero_n2684676970156552555ol_int,B)),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_2208_of__bool__half__eq__0,axiom,
    ! [B: bool] : divide6298287555418463151nteger(zero_n356916108424825756nteger(B),numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_2209_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_2210_one__mod__exp__eq__one,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_2211_even__succ__div__2,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_2212_even__succ__div__2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),A)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_2213_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_2214_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_2215_even__succ__div__two,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_2216_even__succ__div__two,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_2217_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2218_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2219_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2220_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2221_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_2222_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_2223_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_2224_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_2225_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% one_div_2_pow_eq
tff(fact_2226_bits__1__div__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_2227_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_2228_bits__1__div__exp,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% bits_1_div_exp
tff(fact_2229_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_2230_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_2231_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_2232_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(N),K))) ).

% set_bit_greater_eq
tff(fact_2233_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_2234_not__one__le__zero,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).

% not_one_le_zero
tff(fact_2235_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_2236_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_2237_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_2238_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2239_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2240_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2241_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2242_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2243_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_2244_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% zero_less_one_class.zero_le_one
tff(fact_2245_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_2246_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_2247_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_2248_one__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(N))) ).

% one_le_numeral
tff(fact_2249_one__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),numeral_numeral_real(N))) ).

% one_le_numeral
tff(fact_2250_one__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% one_le_numeral
tff(fact_2251_one__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).

% one_le_numeral
tff(fact_2252_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_2253_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_2254_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_2255_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_2256_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_2257_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_real,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,aa_nat_nat(suc,N2))),aa_nat_real(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N3)),aa_nat_real(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_2258_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_2259_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_2260_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_2261_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N: nat,N3: nat] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,N))) ) ) ).

% lift_Suc_antimono_le
tff(fact_2262_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% one_le_power
tff(fact_2263_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% one_le_power
tff(fact_2264_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% one_le_power
tff(fact_2265_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% one_le_power
tff(fact_2266_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).

% one_le_power
tff(fact_2267_le__some__optE,axiom,
    ! [M: real,X: option_real] :
      ( ord_le8614940839814719452n_real(some_real(M),X)
     => ~ ! [M6: real] :
            ( ( X = some_real(M6) )
           => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M),M6)) ) ) ).

% le_some_optE
tff(fact_2268_le__some__optE,axiom,
    ! [M: set_nat,X: option_set_nat] :
      ( ord_le2843612097646854710et_nat(some_set_nat(M),X)
     => ~ ! [M6: set_nat] :
            ( ( X = some_set_nat(M6) )
           => ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(M),M6)) ) ) ).

% le_some_optE
tff(fact_2269_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,some_num(M)),X))
     => ~ ! [M6: num] :
            ( ( X = some_num(M6) )
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),M6)) ) ) ).

% le_some_optE
tff(fact_2270_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,some_nat(M)),X))
     => ~ ! [M6: nat] :
            ( ( X = some_nat(M6) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),M6)) ) ) ).

% le_some_optE
tff(fact_2271_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ord_le1736525451366464988on_int(some_int(M),X)
     => ~ ! [M6: int] :
            ( ( X = some_int(M6) )
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),M6)) ) ) ).

% le_some_optE
tff(fact_2272_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P))) ).

% zero_less_eq_of_bool
tff(fact_2273_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_bool_real(zero_n3304061248610475627l_real,P))) ).

% zero_less_eq_of_bool
tff(fact_2274_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_bool_nat(zero_n2687167440665602831ol_nat,P))) ).

% zero_less_eq_of_bool
tff(fact_2275_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P))) ).

% zero_less_eq_of_bool
tff(fact_2276_zero__less__eq__of__bool,axiom,
    ! [P: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P))) ).

% zero_less_eq_of_bool
tff(fact_2277_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_2278_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_2279_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_2280_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_2281_of__bool__eq__iff,axiom,
    ! [P3: bool,Q2: bool] :
      ( ( aa_bool_nat(zero_n2687167440665602831ol_nat,P3) = aa_bool_nat(zero_n2687167440665602831ol_nat,Q2) )
    <=> ( pp(P3)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_2282_of__bool__eq__iff,axiom,
    ! [P3: bool,Q2: bool] :
      ( ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = aa_bool_int(zero_n2684676970156552555ol_int,Q2) )
    <=> ( pp(P3)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_2283_of__bool__eq__iff,axiom,
    ! [P3: bool,Q2: bool] :
      ( ( zero_n356916108424825756nteger(P3) = zero_n356916108424825756nteger(Q2) )
    <=> ( pp(P3)
      <=> pp(Q2) ) ) ).

% of_bool_eq_iff
tff(fact_2284_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat)) ).

% of_bool_less_eq_one
tff(fact_2285_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(zero_n3304061248610475627l_real,P)),one_one_real)) ).

% of_bool_less_eq_one
tff(fact_2286_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),one_one_nat)) ).

% of_bool_less_eq_one
tff(fact_2287_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int)) ).

% of_bool_less_eq_one
tff(fact_2288_of__bool__less__eq__one,axiom,
    ! [P: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P)),one_one_Code_integer)) ).

% of_bool_less_eq_one
tff(fact_2289_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
    <=> ( X = one_one_assn ) ) ).

% one_reorient
tff(fact_2290_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_2291_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_2292_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_2293_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_2294_power__increasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N4))) ) ) ).

% power_increasing
tff(fact_2295_power__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).

% power_increasing
tff(fact_2296_power__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4))) ) ) ).

% power_increasing
tff(fact_2297_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).

% power_increasing
tff(fact_2298_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).

% power_increasing
tff(fact_2299_split__of__bool__asm,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_complex_bool(P,one_one_complex)) )
          | ( ~ pp(P3)
            & ~ pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool_asm
tff(fact_2300_split__of__bool__asm,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,aa_bool_real(zero_n3304061248610475627l_real,P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_real_bool(P,one_one_real)) )
          | ( ~ pp(P3)
            & ~ pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool_asm
tff(fact_2301_split__of__bool__asm,axiom,
    ! [P: fun_rat_bool,P3: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_rat_bool(P,one_one_rat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool_asm
tff(fact_2302_split__of__bool__asm,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,aa_bool_nat(zero_n2687167440665602831ol_nat,P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_nat_bool(P,one_one_nat)) )
          | ( ~ pp(P3)
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool_asm
tff(fact_2303_split__of__bool__asm,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_int_bool(P,one_one_int)) )
          | ( ~ pp(P3)
            & ~ pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool_asm
tff(fact_2304_split__of__bool__asm,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P3: bool] :
      ( pp(aa_Code_integer_bool(P,zero_n356916108424825756nteger(P3)))
    <=> ~ ( ( pp(P3)
            & ~ pp(aa_Code_integer_bool(P,one_one_Code_integer)) )
          | ( ~ pp(P3)
            & ~ pp(aa_Code_integer_bool(P,zero_z3403309356797280102nteger)) ) ) ) ).

% split_of_bool_asm
tff(fact_2305_split__of__bool,axiom,
    ! [P: fun_complex_bool,P3: bool] :
      ( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P3)))
    <=> ( ( pp(P3)
         => pp(aa_complex_bool(P,one_one_complex)) )
        & ( ~ pp(P3)
         => pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).

% split_of_bool
tff(fact_2306_split__of__bool,axiom,
    ! [P: fun_real_bool,P3: bool] :
      ( pp(aa_real_bool(P,aa_bool_real(zero_n3304061248610475627l_real,P3)))
    <=> ( ( pp(P3)
         => pp(aa_real_bool(P,one_one_real)) )
        & ( ~ pp(P3)
         => pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).

% split_of_bool
tff(fact_2307_split__of__bool,axiom,
    ! [P: fun_rat_bool,P3: bool] :
      ( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P3)))
    <=> ( ( pp(P3)
         => pp(aa_rat_bool(P,one_one_rat)) )
        & ( ~ pp(P3)
         => pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).

% split_of_bool
tff(fact_2308_split__of__bool,axiom,
    ! [P: fun_nat_bool,P3: bool] :
      ( pp(aa_nat_bool(P,aa_bool_nat(zero_n2687167440665602831ol_nat,P3)))
    <=> ( ( pp(P3)
         => pp(aa_nat_bool(P,one_one_nat)) )
        & ( ~ pp(P3)
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_of_bool
tff(fact_2309_split__of__bool,axiom,
    ! [P: fun_int_bool,P3: bool] :
      ( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P3)))
    <=> ( ( pp(P3)
         => pp(aa_int_bool(P,one_one_int)) )
        & ( ~ pp(P3)
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% split_of_bool
tff(fact_2310_split__of__bool,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P3: bool] :
      ( pp(aa_Code_integer_bool(P,zero_n356916108424825756nteger(P3)))
    <=> ( ( pp(P3)
         => pp(aa_Code_integer_bool(P,one_one_Code_integer)) )
        & ( ~ pp(P3)
         => pp(aa_Code_integer_bool(P,zero_z3403309356797280102nteger)) ) ) ) ).

% split_of_bool
tff(fact_2311_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = one_one_complex ) )
      & ( ~ pp(P3)
       => ( zero_n1201886186963655149omplex(P3) = zero_zero_complex ) ) ) ).

% of_bool_def
tff(fact_2312_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( aa_bool_real(zero_n3304061248610475627l_real,P3) = one_one_real ) )
      & ( ~ pp(P3)
       => ( aa_bool_real(zero_n3304061248610475627l_real,P3) = zero_zero_real ) ) ) ).

% of_bool_def
tff(fact_2313_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n2052037380579107095ol_rat(P3) = one_one_rat ) )
      & ( ~ pp(P3)
       => ( zero_n2052037380579107095ol_rat(P3) = zero_zero_rat ) ) ) ).

% of_bool_def
tff(fact_2314_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( aa_bool_nat(zero_n2687167440665602831ol_nat,P3) = one_one_nat ) )
      & ( ~ pp(P3)
       => ( aa_bool_nat(zero_n2687167440665602831ol_nat,P3) = zero_zero_nat ) ) ) ).

% of_bool_def
tff(fact_2315_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = one_one_int ) )
      & ( ~ pp(P3)
       => ( aa_bool_int(zero_n2684676970156552555ol_int,P3) = zero_zero_int ) ) ) ).

% of_bool_def
tff(fact_2316_of__bool__def,axiom,
    ! [P3: bool] :
      ( ( pp(P3)
       => ( zero_n356916108424825756nteger(P3) = one_one_Code_integer ) )
      & ( ~ pp(P3)
       => ( zero_n356916108424825756nteger(P3) = zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
tff(fact_2317_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),X))
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_2318_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_2319_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2320_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2321_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2322_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_decreasing
tff(fact_2323_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_decreasing
tff(fact_2324_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2325_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2326_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2327_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2328_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_le_imp_le_exp
tff(fact_2329_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2330_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2331_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2332_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2333_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2334_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2335_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2336_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2337_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),one_one_Code_integer)) ) ) ) ).

% mult_le_one
tff(fact_2338_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_2339_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_2340_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_2341_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),B)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_2342_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2343_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2344_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2345_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2346_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2347_discrete,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,one_one_Code_integer)),B)) ) ).

% discrete
tff(fact_2348_discrete,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).

% discrete
tff(fact_2349_discrete,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),one_one_int)),B)) ) ).

% discrete
tff(fact_2350_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer)) ) ) ).

% power_le_one
tff(fact_2351_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_2352_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2353_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2354_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2355_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_2356_mod__power__lem,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ( aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),M)) = zero_zero_int ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ( aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),M)) = aa_nat_int(power_power_int(A),N) ) ) ) ) ).

% mod_power_lem
tff(fact_2357_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z)) ) ).

% zless_imp_add1_zle
tff(fact_2358_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% add1_zle_eq
tff(fact_2359_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_real(zero_n3304061248610475627l_real,aa_bool_bool(fconj(P),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_bool_real(zero_n3304061248610475627l_real,P)),aa_bool_real(zero_n3304061248610475627l_real,Q)) ).

% of_bool_conj
tff(fact_2360_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P),Q)) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)) ).

% of_bool_conj
tff(fact_2361_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)) ).

% of_bool_conj
tff(fact_2362_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P),Q)) = aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_conj
tff(fact_2363_of__bool__conj,axiom,
    ! [P: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fconj(P),Q)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)) ).

% of_bool_conj
tff(fact_2364_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(plus_plus_int(aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(plus_plus_int(K),L)) ).

% signed_take_bit_add
tff(fact_2365_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(times_times_int(K),L)) ).

% signed_take_bit_mult
tff(fact_2366_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_2367_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_2368_le__numeral__extra_I3_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% le_numeral_extra(3)
tff(fact_2369_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_2370_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_2371_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_2372_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X3)) ) ).

% minf(8)
tff(fact_2373_minf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,T),X3)) ) ).

% minf(8)
tff(fact_2374_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X3)) ) ).

% minf(8)
tff(fact_2375_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X3)) ) ).

% minf(8)
tff(fact_2376_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X3)) ) ).

% minf(8)
tff(fact_2377_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3)) ) ).

% minf(8)
tff(fact_2378_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z5))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),T)) ) ).

% minf(6)
tff(fact_2379_minf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),Z5))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),T)) ) ).

% minf(6)
tff(fact_2380_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z5))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),T)) ) ).

% minf(6)
tff(fact_2381_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z5))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),T)) ) ).

% minf(6)
tff(fact_2382_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z5))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),T)) ) ).

% minf(6)
tff(fact_2383_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z5))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T)) ) ).

% minf(6)
tff(fact_2384_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X3)) ) ).

% pinf(8)
tff(fact_2385_pinf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,T),X3)) ) ).

% pinf(8)
tff(fact_2386_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X3)) ) ).

% pinf(8)
tff(fact_2387_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X3)) ) ).

% pinf(8)
tff(fact_2388_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X3)) ) ).

% pinf(8)
tff(fact_2389_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3)) ) ).

% pinf(8)
tff(fact_2390_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),X3))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),T)) ) ).

% pinf(6)
tff(fact_2391_pinf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z5: code_integer] :
    ! [X3: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z5),X3))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X3),T)) ) ).

% pinf(6)
tff(fact_2392_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),X3))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),T)) ) ).

% pinf(6)
tff(fact_2393_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X3: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z5),X3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),T)) ) ).

% pinf(6)
tff(fact_2394_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z5),X3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),T)) ) ).

% pinf(6)
tff(fact_2395_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),X3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T)) ) ).

% pinf(6)
tff(fact_2396_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A3: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A3))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_2397_verit__comp__simplify1_I3_J,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B3),A3))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_2398_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A3: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A3))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_2399_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A3: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A3))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_2400_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A3: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A3))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_2401_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A3: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3)) ) ).

% verit_comp_simplify1(3)
tff(fact_2402_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2403_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2404_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2405_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_2406_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2407_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2408_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2409_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_2410_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_2411_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))) ) ).

% add_right_mono
tff(fact_2412_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_right_mono
tff(fact_2413_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_right_mono
tff(fact_2414_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))) ) ).

% add_right_mono
tff(fact_2415_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_2416_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))) ) ).

% add_left_mono
tff(fact_2417_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_left_mono
tff(fact_2418_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_left_mono
tff(fact_2419_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))) ) ).

% add_left_mono
tff(fact_2420_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_mono
tff(fact_2421_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_mono
tff(fact_2422_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_2423_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_mono
tff(fact_2424_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2425_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2426_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2427_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2428_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2429_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2430_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2431_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2432_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J2))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2433_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J2))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2434_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2435_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2436_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_2437_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_leD
tff(fact_2438_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_2439_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).

% le_SucI
tff(fact_2440_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M7))
     => ? [M3: nat] : M7 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_2441_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_2442_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_2443_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_2444_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N2))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N2)) )
     => pp(aa_nat_bool(P,N)) ) ).

% full_nat_induct
tff(fact_2445_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_2446_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R3: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ! [X2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X2),X2))
       => ( ! [X2: nat,Y3: nat,Z5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X2),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y3),Z5))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X2),Z5)) ) )
         => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,N2),aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2447_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2448_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2449_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2450_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2451_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2452_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_2453_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

% less_eq_nat.simps(1)
tff(fact_2454_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_2455_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_2456_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_2457_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_2458_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_2459_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_2460_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_2461_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer)) ).

% less_numeral_extra(4)
tff(fact_2462_exists__leI,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ( ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),N))
           => ~ pp(aa_nat_bool(P,N5)) )
       => pp(aa_nat_bool(P,N)) )
     => ? [N6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N6),N))
          & pp(aa_nat_bool(P,N6)) ) ) ).

% exists_leI
tff(fact_2463_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_2464_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_2465_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_2466_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_2467_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_2468_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J2: nat] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),J))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J2))) ) ) ).

% less_mono_imp_le_mono
tff(fact_2469_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(one_one_assn),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2470_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2471_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2472_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2473_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2474_mult_Ocomm__neutral,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(A),one_one_assn) = A ).

% mult.comm_neutral
tff(fact_2475_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_2476_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2477_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_2478_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_2479_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% add_leE
tff(fact_2480_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

% le_add1
tff(fact_2481_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% le_add2
tff(fact_2482_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% add_leD1
tff(fact_2483_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% add_leD2
tff(fact_2484_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_2485_add__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ) ).

% add_le_mono
tff(fact_2486_add__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K))) ) ).

% add_le_mono1
tff(fact_2487_trans__le__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M))) ) ).

% trans_le_add1
tff(fact_2488_trans__le__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J2))) ) ).

% trans_le_add2
tff(fact_2489_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_2490_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_2491_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_2492_imp__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_2493_conj__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_2494_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2495_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_2496_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).

% div_le_mono
tff(fact_2497_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).

% div_le_dividend
tff(fact_2498_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_2499_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_2500_mult__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),L))) ) ) ).

% mult_le_mono
tff(fact_2501_mult__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K))) ) ).

% mult_le_mono1
tff(fact_2502_mult__le__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J2))) ) ).

% mult_le_mono2
tff(fact_2503_one__dvd,axiom,
    ! [A: assn] : dvd_dvd_assn(one_one_assn,A) ).

% one_dvd
tff(fact_2504_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_2505_one__dvd,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A)) ).

% one_dvd
tff(fact_2506_one__dvd,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).

% one_dvd
tff(fact_2507_one__dvd,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).

% one_dvd
tff(fact_2508_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% unit_imp_dvd
tff(fact_2509_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% unit_imp_dvd
tff(fact_2510_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_2511_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_2512_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).

% mod_less_eq_dividend
tff(fact_2513_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_2514_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_2515_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),zero_z5237406670263579293d_enat))
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_2516_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N)) ).

% i0_lb
tff(fact_2517_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_2518_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2519_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2520_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2521_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(times_times_int(C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2522_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2523_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2524_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2525_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2526_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2527_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2528_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2529_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(times_times_int(B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2530_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2531_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2532_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2533_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2534_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2535_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2536_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2537_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(times_times_int(C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2538_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2539_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2540_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2541_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2542_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2543_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),one_one_Code_integer)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2544_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2545_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(times_times_int(B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2546_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2547_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2548_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2549_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2550_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z5))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z5),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z5),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2551_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z5))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z5),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2552_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V))
           => ( ( plus_p5714425477246183910nteger(U,V) = one_one_Code_integer )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(U),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2553_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),X),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2554_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2555_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),X)),aa_int_int(times_times_int(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2556_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2557_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2558_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2559_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_2560_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2561_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2562_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2563_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2564_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2565_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).

% self_le_power
tff(fact_2566_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% self_le_power
tff(fact_2567_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% self_le_power
tff(fact_2568_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% self_le_power
tff(fact_2569_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).

% self_le_power
tff(fact_2570_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_2571_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),N)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_2572_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)))
      <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),one_one_Code_integer))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% dvd_power_iff
tff(fact_2573_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),K)) ).

% unset_bit_less_eq
tff(fact_2574_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),Z))) ) ).

% le_imp_0_less
tff(fact_2575_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_2576_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2577_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2578_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),X),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2579_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V))
           => ( ( plus_p5714425477246183910nteger(U,V) = one_one_Code_integer )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(U),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2580_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2581_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),X)),aa_int_int(times_times_int(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2582_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N))) ).

% two_realpow_ge_one
tff(fact_2583_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),numera6620942414471956472nteger(N))) ).

% zero_le_numeral
tff(fact_2584_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(N))) ).

% zero_le_numeral
tff(fact_2585_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(N))) ).

% zero_le_numeral
tff(fact_2586_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).

% zero_le_numeral
tff(fact_2587_zero__le__numeral,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).

% zero_le_numeral
tff(fact_2588_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(N)),zero_z3403309356797280102nteger)) ).

% not_numeral_le_zero
tff(fact_2589_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_2590_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_2591_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_2592_not__numeral__le__zero,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_2593_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),B)) ) ) ).

% add_decreasing
tff(fact_2594_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,C)),B)) ) ) ).

% add_decreasing
tff(fact_2595_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2596_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2597_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),B)) ) ) ).

% add_decreasing
tff(fact_2598_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),plus_plus_rat(A,C))) ) ) ).

% add_increasing
tff(fact_2599_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),plus_p5714425477246183910nteger(A,C))) ) ) ).

% add_increasing
tff(fact_2600_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing
tff(fact_2601_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing
tff(fact_2602_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(plus_plus_int(A),C))) ) ) ).

% add_increasing
tff(fact_2603_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),B)) ) ) ).

% add_decreasing2
tff(fact_2604_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,C)),B)) ) ) ).

% add_decreasing2
tff(fact_2605_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2606_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2607_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_2608_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),plus_plus_rat(A,C))) ) ) ).

% add_increasing2
tff(fact_2609_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),plus_p5714425477246183910nteger(A,C))) ) ) ).

% add_increasing2
tff(fact_2610_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing2
tff(fact_2611_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing2
tff(fact_2612_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(plus_plus_int(A),C))) ) ) ).

% add_increasing2
tff(fact_2613_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2614_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2615_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2616_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2617_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_2618_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_2619_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger)) ) ) ).

% add_nonpos_nonpos
tff(fact_2620_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_2621_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_2622_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_2623_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_2624_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( ( plus_p5714425477246183910nteger(X,Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2625_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2626_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2627_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(plus_plus_int(X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2628_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_2629_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),zero_z3403309356797280102nteger))
       => ( ( plus_p5714425477246183910nteger(X,Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2630_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2631_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2632_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(plus_plus_int(X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2633_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2634_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2635_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2636_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2637_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_2638_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2639_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2640_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2641_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2642_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_2643_zero__le__square,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A))) ).

% zero_le_square
tff(fact_2644_zero__le__square,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A))) ).

% zero_le_square
tff(fact_2645_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).

% zero_le_square
tff(fact_2646_zero__le__square,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),A))) ).

% zero_le_square
tff(fact_2647_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))) ) ).

% split_mult_pos_le
tff(fact_2648_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ).

% split_mult_pos_le
tff(fact_2649_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ).

% split_mult_pos_le
tff(fact_2650_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B))) ) ).

% split_mult_pos_le
tff(fact_2651_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2652_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2653_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2654_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_2655_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2656_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2657_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2658_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_2659_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ).

% mult_left_mono
tff(fact_2660_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono
tff(fact_2661_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono
tff(fact_2662_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_left_mono
tff(fact_2663_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% mult_left_mono
tff(fact_2664_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2665_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2666_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2667_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_2668_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))) ) ) ).

% mult_right_mono
tff(fact_2669_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono
tff(fact_2670_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono
tff(fact_2671_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_right_mono
tff(fact_2672_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))) ) ) ).

% mult_right_mono
tff(fact_2673_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2674_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2675_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2676_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_2677_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)) ) ).

% split_mult_neg_le
tff(fact_2678_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_2679_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_2680_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_2681_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_2682_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2683_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2684_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2685_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2686_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_2687_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2688_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2689_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2690_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2691_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_2692_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2693_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2694_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2695_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2696_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_2697_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)),zero_z3403309356797280102nteger)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2698_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2699_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2700_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2701_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(B),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2702_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2703_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2704_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2705_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_2706_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2707_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2708_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2709_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2710_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2711_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2712_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),J2))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J2,L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2713_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2714_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2715_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2716_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J2,L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2717_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J2))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(I,K)),plus_p5714425477246183910nteger(J2,L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2718_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2719_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2720_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2721_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_le_less_mono
tff(fact_2722_add__le__less__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,D))) ) ) ).

% add_le_less_mono
tff(fact_2723_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_le_less_mono
tff(fact_2724_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_2725_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_le_less_mono
tff(fact_2726_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).

% add_less_le_mono
tff(fact_2727_add__less__le__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,C)),plus_p5714425477246183910nteger(B,D))) ) ) ).

% add_less_le_mono
tff(fact_2728_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_less_le_mono
tff(fact_2729_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_2730_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D))) ) ) ).

% add_less_le_mono
tff(fact_2731_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B2),N)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_2732_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2733_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_2734_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).

% divide_right_mono
tff(fact_2735_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_right_mono
tff(fact_2736_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2737_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_2738_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2739_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_2740_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2741_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_2742_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2743_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_2744_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2745_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_2746_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C))) ) ) ).

% divide_right_mono_neg
tff(fact_2747_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_2748_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).

% power_mono
tff(fact_2749_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono
tff(fact_2750_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono
tff(fact_2751_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).

% power_mono
tff(fact_2752_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono
tff(fact_2753_zero__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% zero_le_power
tff(fact_2754_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_power
tff(fact_2755_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_le_power
tff(fact_2756_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% zero_le_power
tff(fact_2757_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_power
tff(fact_2758_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2759_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2760_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2761_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_2762_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_2763_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_2764_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_2765_zero__less__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% zero_less_one
tff(fact_2766_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_2767_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_2768_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_2769_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_2770_not__one__less__zero,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).

% not_one_less_zero
tff(fact_2771_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_2772_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_2773_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_2774_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_2775_less__numeral__extra_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% less_numeral_extra(1)
tff(fact_2776_nat__compl__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N2))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct
tff(fact_2777_nat__compl__induct_H,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N2))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct'
tff(fact_2778_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),one_one_Code_integer)) ).

% not_numeral_less_one
tff(fact_2779_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),one_one_real)) ).

% not_numeral_less_one
tff(fact_2780_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_2781_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_2782_not__numeral__less__one,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).

% not_numeral_less_one
tff(fact_2783_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(suc,A))) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_2784_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(suc,A))) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_2785_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).

% Suc_leI
tff(fact_2786_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_eq
tff(fact_2787_dec__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J2))
               => ( pp(aa_nat_bool(P,N2))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
         => pp(aa_nat_bool(P,J2)) ) ) ) ).

% dec_induct
tff(fact_2788_inc__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(P,J2))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J2))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
                 => pp(aa_nat_bool(P,N2)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_2789_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_2790_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_2791_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_2792_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_2793_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_2794_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).

% less_add_one
tff(fact_2795_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(A,one_one_rat))) ).

% less_add_one
tff(fact_2796_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_2797_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(plus_plus_int(A),one_one_int))) ).

% less_add_one
tff(fact_2798_less__add__one,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(A,one_one_Code_integer))) ).

% less_add_one
tff(fact_2799_add__mono1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real))) ) ).

% add_mono1
tff(fact_2800_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,one_one_rat)),plus_plus_rat(B,one_one_rat))) ) ).

% add_mono1
tff(fact_2801_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat))) ) ).

% add_mono1
tff(fact_2802_add__mono1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_int_int(plus_plus_int(B),one_one_int))) ) ).

% add_mono1
tff(fact_2803_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,one_one_Code_integer)),plus_p5714425477246183910nteger(B,one_one_Code_integer))) ) ).

% add_mono1
tff(fact_2804_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_2805_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2806_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
tff(fact_2807_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_2808_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_2809_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_2810_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_2811_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_2812_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_2813_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_2814_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_2815_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N))) ) ) ).

% less_1_mult
tff(fact_2816_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N))) ) ) ).

% less_1_mult
tff(fact_2817_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

% less_1_mult
tff(fact_2818_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(times_times_int(M),N))) ) ) ).

% less_1_mult
tff(fact_2819_less__1__mult,axiom,
    ! [M: code_integer,N: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(M),N))) ) ) ).

% less_1_mult
tff(fact_2820_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2821_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2822_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
tff(fact_2823_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_2824_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% le_imp_power_dvd
tff(fact_2825_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% le_imp_power_dvd
tff(fact_2826_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N))) ) ).

% le_imp_power_dvd
tff(fact_2827_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N))) ) ).

% le_imp_power_dvd
tff(fact_2828_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% le_imp_power_dvd
tff(fact_2829_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N))) ) ).

% le_imp_power_dvd
tff(fact_2830_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_2831_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_2832_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_2833_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_2834_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_2835_power__le__dvd,axiom,
    ! [A: rat,N: nat,B: rat,M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),N)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_nat_rat(power_power_rat(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_2836_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M))) ) ) ).

% dvd_power_le
tff(fact_2837_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M))) ) ) ).

% dvd_power_le
tff(fact_2838_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),M))) ) ) ).

% dvd_power_le
tff(fact_2839_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).

% dvd_power_le
tff(fact_2840_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M))) ) ) ).

% dvd_power_le
tff(fact_2841_dvd__power__le,axiom,
    ! [X: rat,Y: rat,N: nat,M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),M))) ) ) ).

% dvd_power_le
tff(fact_2842_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_2843_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(times_times_assn(X),Y) = one_one_assn )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(X),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_2844_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_2845_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_2846_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_2847_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2848_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2849_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(times_times_int(X),Y) = one_one_int )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2850_not__is__unit__0,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).

% not_is_unit_0
tff(fact_2851_not__is__unit__0,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).

% not_is_unit_0
tff(fact_2852_not__is__unit__0,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% not_is_unit_0
tff(fact_2853_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_2854_power__one__over,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),N) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),N)) ).

% power_one_over
tff(fact_2855_power__one__over,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_one_over
tff(fact_2856_power__one__over,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),N) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),N)) ).

% power_one_over
tff(fact_2857_power__0,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_2858_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2859_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2860_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2861_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2862_power__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_2863_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2864_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_2865_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_2866_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_2867_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_2868_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_2869_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_2870_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2871_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2872_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2873_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2874_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2875_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2876_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2877_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(times_times_int(B),A) = aa_int_int(times_times_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2878_mlex__leI,axiom,
    ! [A: nat,A3: nat,B: nat,B3: nat,N4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),B3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N4)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N4)),B3))) ) ) ).

% mlex_leI
tff(fact_2879_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_2880_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_2881_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_2882_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_2883_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_2884_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_2885_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_2886_int__mod__ge,axiom,
    ! [A: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(modulo_modulo_int(A),N))) ) ) ).

% int_mod_ge
tff(fact_2887_int__mod__eq,axiom,
    ! [B: int,N: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),N))
       => ( ( aa_int_int(modulo_modulo_int(A),N) = aa_int_int(modulo_modulo_int(B),N) )
         => ( aa_int_int(modulo_modulo_int(A),N) = B ) ) ) ) ).

% int_mod_eq
tff(fact_2888_int__mod__lem,axiom,
    ! [N: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),N)) )
      <=> ( aa_int_int(modulo_modulo_int(B),N) = B ) ) ) ).

% int_mod_lem
tff(fact_2889_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_2890_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2891_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B))) ) ) ).

% neg_mod_conj
tff(fact_2892_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_2893_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_2894_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_2895_div__mult__le,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),A)) ).

% div_mult_le
tff(fact_2896_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_2897_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_2898_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_2899_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_2900_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2901_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_2902_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2903_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_2904_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).

% nonneg_mod_div
tff(fact_2905_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2906_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_2907_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2))) ) ).

% real_arch_pow
tff(fact_2908_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_2909_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(plus_plus_int(K),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I3))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_2910_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(pow,X),one) = X ).

% pow.simps(1)
tff(fact_2911_emep1,axiom,
    ! [N: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
         => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(N),one_one_int)),D) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(N),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_2912_eme1p,axiom,
    ! [N: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
         => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),N)),D) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(N),D)) ) ) ) ) ).

% eme1p
tff(fact_2913_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ).

% verit_le_mono_div
tff(fact_2914_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2915_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2916_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_2917_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger)) ) ) ).

% add_neg_nonpos
tff(fact_2918_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_2919_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_2920_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_2921_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).

% add_nonneg_pos
tff(fact_2922_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B))) ) ) ).

% add_nonneg_pos
tff(fact_2923_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2924_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2925_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B))) ) ) ).

% add_nonneg_pos
tff(fact_2926_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_2927_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),zero_z3403309356797280102nteger)) ) ) ).

% add_nonpos_neg
tff(fact_2928_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_2929_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_2930_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_2931_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).

% add_pos_nonneg
tff(fact_2932_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(A,B))) ) ) ).

% add_pos_nonneg
tff(fact_2933_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2934_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2935_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B))) ) ) ).

% add_pos_nonneg
tff(fact_2936_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),plus_plus_rat(A,C))) ) ) ).

% add_strict_increasing
tff(fact_2937_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),plus_p5714425477246183910nteger(A,C))) ) ) ).

% add_strict_increasing
tff(fact_2938_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2939_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_2940_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(plus_plus_int(A),C))) ) ) ).

% add_strict_increasing
tff(fact_2941_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),plus_plus_rat(A,C))) ) ) ).

% add_strict_increasing2
tff(fact_2942_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),plus_p5714425477246183910nteger(A,C))) ) ) ).

% add_strict_increasing2
tff(fact_2943_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2944_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2945_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(plus_plus_int(A),C))) ) ) ).

% add_strict_increasing2
tff(fact_2946_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(Y,E2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_2947_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_2948_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2949_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),Z)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Z)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2950_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2951_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(X),Z)),aa_int_int(times_times_int(Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2952_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),aa_rat_rat(times_times_rat(Z),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2953_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),X)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),Y)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2954_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2955_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(Z),X)),aa_int_int(times_times_int(Z),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2956_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2957_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2958_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2959_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2960_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2961_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2962_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2963_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2964_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2965_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2966_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2967_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2968_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2969_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2970_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2971_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2972_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2973_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2974_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2975_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2976_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2977_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2978_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2979_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2980_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2981_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2982_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2983_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2984_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2985_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2986_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2987_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2988_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2989_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2990_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2991_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2992_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2993_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2994_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2995_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2996_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2997_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2998_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2999_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3000_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3001_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3002_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3003_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3004_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_3005_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3006_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3007_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3008_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3009_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_3010_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3011_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3012_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3013_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3014_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3015_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3016_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3017_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3018_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3019_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3020_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3021_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3022_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3023_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_3024_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y))),zero_z3403309356797280102nteger))
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3025_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3026_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3027_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3028_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3029_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).

% frac_le
tff(fact_3030_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).

% frac_le
tff(fact_3031_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).

% frac_less
tff(fact_3032_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).

% frac_less
tff(fact_3033_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Z))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).

% frac_less2
tff(fact_3034_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Z))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).

% frac_less2
tff(fact_3035_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_3036_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_3037_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_3038_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_3039_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).

% divide_nonneg_pos
tff(fact_3040_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_3041_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).

% divide_nonpos_neg
tff(fact_3042_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_3043_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_3044_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_3045_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3046_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3047_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3048_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),divide6298287555418463151nteger(A,B))) ) ) ).

% div_positive
tff(fact_3049_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) ) ) ).

% div_positive
tff(fact_3050_div__positive,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% div_positive
tff(fact_3051_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3052_power__less__imp__less__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3053_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3054_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3055_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_3056_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3057_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3058_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3059_power__inject__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3060_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3061_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3062_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3063_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3064_power__le__imp__le__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3065_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3066_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3067_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3068_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_3069_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3070_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3071_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3072_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3073_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3074_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3075_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_3076_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,one_one_rat))) ).

% zero_less_two
tff(fact_3077_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_3078_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_3079_zero__less__two,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,one_one_Code_integer))) ).

% zero_less_two
tff(fact_3080_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_3081_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3082_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3083_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_3084_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),divide_divide_rat(B,A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_3085_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(B,A),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_3086_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3087_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3088_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),B) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,B),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_3089_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3090_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3091_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B)) ) ).

% gt_half_sum
tff(fact_3092_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat))),B)) ) ).

% gt_half_sum
tff(fact_3093_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_3094_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat)))) ) ).

% less_half_sum
tff(fact_3095_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_gt1_lemma
tff(fact_3096_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3097_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_gt1_lemma
tff(fact_3098_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3099_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))) ) ).

% power_gt1_lemma
tff(fact_3100_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).

% power_less_power_Suc
tff(fact_3101_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3102_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).

% power_less_power_Suc
tff(fact_3103_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3104_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))) ) ).

% power_less_power_Suc
tff(fact_3105_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3106_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3107_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3108_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3109_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N)))) ) ).

% power_gt1
tff(fact_3110_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_3111_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_3112_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_3113_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_3114_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
tff(fact_3115_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_3116_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C2) ) ) ).

% unit_dvdE
tff(fact_3117_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_3118_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(times_times_int(A),C2) ) ) ).

% unit_dvdE
tff(fact_3119_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_3120_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_3121_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

% nat_one_le_power
tff(fact_3122_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3123_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3124_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3125_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3126_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).

% power_less_imp_less_exp
tff(fact_3127_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4))) ) ) ).

% power_strict_increasing
tff(fact_3128_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_3129_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4))) ) ) ).

% power_strict_increasing
tff(fact_3130_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_3131_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N4))) ) ) ).

% power_strict_increasing
tff(fact_3132_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_3133_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3134_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3135_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3136_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3137_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(times_times_int(C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3138_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3139_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(times_times_int(A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3140_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3141_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3142_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3143_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3144_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3145_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( aa_int_int(times_times_int(A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3146_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3147_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3148_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_3149_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_3150_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3151_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_3152_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3153_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3154_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3155_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).

% mod_le_divisor
tff(fact_3156_int__mod__ge_H,axiom,
    ! [B: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(B),N)),aa_int_int(modulo_modulo_int(B),N))) ) ) ).

% int_mod_ge'
tff(fact_3157_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(plus_plus_int(K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3158_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(P,N2))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_3159_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q3: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3160_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_3161_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_3162_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),A)) ) ) ).

% zdiv_le_dividend
tff(fact_3163_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B))) ) ) ).

% zdiv_mono1
tff(fact_3164_zdiv__mono2,axiom,
    ! [A: int,B3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3))) ) ) ) ).

% zdiv_mono2
tff(fact_3165_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3166_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_3167_zdiv__mono2__neg,axiom,
    ! [A: int,B3: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3168_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_3169_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).

% div_positive_int
tff(fact_3170_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_3171_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_3172_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3173_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3174_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3175_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3176_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3177_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_3178_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_3179_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_3180_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(times_times_int(M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3181_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_3182_power__2__mult__step__le,axiom,
    ! [N3: nat,N: nat,K4: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N3)),K4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),K)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K4),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),K))) ) ) ).

% power_2_mult_step_le
tff(fact_3183_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3184_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3185_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_3186_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_3187_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_3188_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_3189_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_3190_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_left_mono
tff(fact_3191_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono
tff(fact_3192_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3193_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_3194_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_3195_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_3196_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_3197_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_3198_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3199_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_3200_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3201_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).

% mult_imp_div_pos_le
tff(fact_3202_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),divide_divide_rat(X,Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3203_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_3204_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3205_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_3206_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
         => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3207_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3208_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3209_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3210_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3211_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3212_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3213_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3214_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3215_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3216_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_3217_pos__mod__sign2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% pos_mod_sign2
tff(fact_3218_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% power_Suc_less
tff(fact_3219_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% power_Suc_less
tff(fact_3220_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% power_Suc_less
tff(fact_3221_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).

% power_Suc_less
tff(fact_3222_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).

% power_Suc_less
tff(fact_3223_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_3224_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_3225_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_3226_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_3227_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_3228_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_3229_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_3230_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N))),one_one_Code_integer)) ) ) ).

% power_Suc_less_one
tff(fact_3231_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3232_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N4)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3233_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N4)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3234_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N4)),aa_nat_int(power_power_int(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3235_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N4)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ) ).

% power_strict_decreasing
tff(fact_3236_odd__one,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),one_one_nat)) ).

% odd_one
tff(fact_3237_odd__one,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),one_one_int)) ).

% odd_one
tff(fact_3238_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ).

% one_power2
tff(fact_3239_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_real ).

% one_power2
tff(fact_3240_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_int ).

% one_power2
tff(fact_3241_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_complex ).

% one_power2
tff(fact_3242_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_Code_integer ).

% one_power2
tff(fact_3243_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_rat ).

% one_power2
tff(fact_3244_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B4),one_one_Code_integer))
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B4 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B4) = A )
                   => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B4) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3245_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B4),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3246_is__unitE,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B4),one_one_int))
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
                   => ( ( aa_int_int(times_times_int(A),B4) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(times_times_int(C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3247_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3248_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3249_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3250_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3251_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3252_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3253_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).

% one_less_power
tff(fact_3254_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).

% one_less_power
tff(fact_3255_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).

% one_less_power
tff(fact_3256_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).

% one_less_power
tff(fact_3257_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ) ).

% one_less_power
tff(fact_3258_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q2))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_3259_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C))) ) ) ).

% td_gal
tff(fact_3260_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3261_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))) ) ).

% dvd_power
tff(fact_3262_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ).

% dvd_power
tff(fact_3263_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).

% dvd_power
tff(fact_3264_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N))) ) ).

% dvd_power
tff(fact_3265_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_Code_integer ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N))) ) ).

% dvd_power
tff(fact_3266_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        | ( X = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N))) ) ).

% dvd_power
tff(fact_3267_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_3268_nmod2,axiom,
    ! [N: int] :
      ( ( aa_int_int(modulo_modulo_int(N),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int )
      | ( aa_int_int(modulo_modulo_int(N),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% nmod2
tff(fact_3269_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% nat_1_add_1
tff(fact_3270_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q6)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_3271_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q6: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q6)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3272_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q6: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q6)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q6)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3273_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q6)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q6),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3274_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q6)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3275_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X2: int] :
            ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X3: int] :
              ( pp(aa_int_bool(P,X3))
             => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X3),aa_int_int(times_times_int(K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3276_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3277_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3278_split__zmod,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(modulo_modulo_int(N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I2: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I2: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_3279_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3280_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3281_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3282_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))))
         => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3283_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3284_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3285_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% of_bool_odd_eq_mod_2
tff(fact_3286_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% of_bool_odd_eq_mod_2
tff(fact_3287_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% of_bool_odd_eq_mod_2
tff(fact_3288_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri631733984087533419it_int(M),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)) ) ).

% even_signed_take_bit_iff
tff(fact_3289_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3290_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3291_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3292_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3293_zero__le__power2,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% zero_le_power2
tff(fact_3294_zero__le__power2,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% zero_le_power2
tff(fact_3295_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% zero_le_power2
tff(fact_3296_zero__le__power2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% zero_le_power2
tff(fact_3297_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3298_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3299_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3300_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3301_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3302_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3303_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3304_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3305_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3306_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3307_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_3308_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_3309_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_3310_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).

% power_strict_mono
tff(fact_3311_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).

% power_strict_mono
tff(fact_3312_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3313_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3314_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3315_power__mono__odd,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).

% power_mono_odd
tff(fact_3316_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_odd
tff(fact_3317_power__mono__odd,axiom,
    ! [N: nat,A: real,B: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_odd
tff(fact_3318_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_odd
tff(fact_3319_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q4))))
            & pp(aa_nat_bool(P,Q4)) ) ) ) ).

% split_div'
tff(fact_3320_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_3321_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3322_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3323_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3324_nat__induct2,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N2: nat] :
              ( pp(aa_nat_bool(P,N2))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct2
tff(fact_3325_axxmod2,axiom,
    ! [X: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),X)),X)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int )
      & ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(zero_zero_int),X)),X)),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_3326_split__zdiv,axiom,
    ! [P: fun_int_bool,N: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I2: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J3) ) )
             => pp(aa_int_bool(P,I2)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I2: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J3) ) )
             => pp(aa_int_bool(P,I2)) ) ) ) ) ).

% split_zdiv
tff(fact_3327_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3328_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3329_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I2: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_3330_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(modulo_modulo_int(N),K)))
      <=> ! [I2: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
              & ( N = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I2),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_3331_z1pmod2,axiom,
    ! [B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% z1pmod2
tff(fact_3332_p1mod22k,axiom,
    ! [B: int,N: nat] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),one_one_int)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(B),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))),one_one_int) ).

% p1mod22k
tff(fact_3333_p1mod22k_H,axiom,
    ! [B: int,N: nat] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(B),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))) ).

% p1mod22k'
tff(fact_3334_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3335_bits__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),aa_num_nat(numeral_numeral_nat,bit0(one))) = A4 )
         => pp(aa_nat_bool(P,A4)) )
     => ( ! [A4: nat,B4: bool] :
            ( pp(aa_nat_bool(P,A4))
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A4))),aa_num_nat(numeral_numeral_nat,bit0(one))) = A4 )
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,B4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A4)))) ) )
       => pp(aa_nat_bool(P,A)) ) ) ).

% bits_induct
tff(fact_3336_bits__induct,axiom,
    ! [P: fun_int_bool,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),aa_num_int(numeral_numeral_int,bit0(one))) = A4 )
         => pp(aa_int_bool(P,A4)) )
     => ( ! [A4: int,B4: bool] :
            ( pp(aa_int_bool(P,A4))
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,B4)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A4))),aa_num_int(numeral_numeral_int,bit0(one))) = A4 )
             => pp(aa_int_bool(P,aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,B4)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A4)))) ) )
       => pp(aa_int_bool(P,A)) ) ) ).

% bits_induct
tff(fact_3337_bits__induct,axiom,
    ! [P: fun_Co8822979462182099027r_bool,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(bit0(one))) = A4 )
         => pp(aa_Code_integer_bool(P,A4)) )
     => ( ! [A4: code_integer,B4: bool] :
            ( pp(aa_Code_integer_bool(P,A4))
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B4),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),A4)),numera6620942414471956472nteger(bit0(one))) = A4 )
             => pp(aa_Code_integer_bool(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B4),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),A4)))) ) )
       => pp(aa_Code_integer_bool(P,A)) ) ) ).

% bits_induct
tff(fact_3338_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3339_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3340_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3341_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3342_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3343_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_power2_ge_zero
tff(fact_3344_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_power2_ge_zero
tff(fact_3345_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_power2_ge_zero
tff(fact_3346_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_power2_ge_zero
tff(fact_3347_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_z3403309356797280102nteger))
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3348_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3349_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3350_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3351_zero__le__even__power_H,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% zero_le_even_power'
tff(fact_3352_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% zero_le_even_power'
tff(fact_3353_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% zero_le_even_power'
tff(fact_3354_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% zero_le_even_power'
tff(fact_3355_zero__le__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3356_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3357_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3358_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_3359_zero__le__odd__power,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ) ).

% zero_le_odd_power
tff(fact_3360_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% zero_le_odd_power
tff(fact_3361_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_3362_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_3363_zero__le__even__power,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).

% zero_le_even_power
tff(fact_3364_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).

% zero_le_even_power
tff(fact_3365_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).

% zero_le_even_power
tff(fact_3366_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).

% zero_le_even_power
tff(fact_3367_oddE,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B4)),one_one_nat) ) ).

% oddE
tff(fact_3368_oddE,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
     => ~ ! [B4: int] : A != aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B4)),one_one_int) ) ).

% oddE
tff(fact_3369_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% two_pow_div_gt_le
tff(fact_3370_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_3371_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_3372_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_3373_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != zero_zero_nat ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3374_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3375_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A))
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_3376_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_3377_axxdiv2,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),X)),X)),aa_num_int(numeral_numeral_int,bit0(one))) = X )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(zero_zero_int),X)),X)),aa_num_int(numeral_numeral_int,bit0(one))) = X ) ) ).

% axxdiv2
tff(fact_3378_z1pdiv2,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = B ).

% z1pdiv2
tff(fact_3379_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ).

% exp_mod_exp
tff(fact_3380_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) ).

% exp_mod_exp
tff(fact_3381_exp__mod__exp,axiom,
    ! [M: nat,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)) ).

% exp_mod_exp
tff(fact_3382_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),X)),Y)),plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_squares_bound
tff(fact_3383_sum__squares__bound,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sum_squares_bound
tff(fact_3384_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3385_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3386_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3387_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3388_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K))))) ) ).

% sb_inc_lem
tff(fact_3389_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3390_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3391_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( ( aa_int_int(modulo_modulo_int(X),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) = aa_int_int(modulo_modulo_int(X),M) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3392_power__le__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),zero_z3403309356797280102nteger))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3393_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3394_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3395_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3396_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_3397_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_rat_rat(times_times_rat(X),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),divide_divide_rat(plus_plus_rat(X,Y),numeral_numeral_rat(bit0(one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3398_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(bit0(one))))) ) ) ) ).

% arith_geo_mean
tff(fact_3399_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_or4029947393144176647an_rat(L,U)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3400_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or66887138388493659n_real(L,U)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3401_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or3540276404033026485et_nat(L,U))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(L),I))
        & pp(aa_set_nat_bool(ord_less_set_nat(I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3402_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( pp(aa_set_num_bool(member_num(I),set_or1222409239386451017an_num(L,U)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3403_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3404_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or4662586982721622107an_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3405_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),set_or8404916559141939852nteger(L,U)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,L),I))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),U)) ) ) ).

% atLeastLessThan_iff
tff(fact_3406_ivl__subset,axiom,
    ! [I: real,J2: real,M: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or66887138388493659n_real(I,J2)),set_or66887138388493659n_real(M,N)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,J2),I))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M),I))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,J2),N)) ) ) ) ).

% ivl_subset
tff(fact_3407_ivl__subset,axiom,
    ! [I: num,J2: num,M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(I,J2)),set_or1222409239386451017an_num(M,N)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,J2),I))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),I))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,J2),N)) ) ) ) ).

% ivl_subset
tff(fact_3408_ivl__subset,axiom,
    ! [I: nat,J2: nat,M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J2)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),N)) ) ) ) ).

% ivl_subset
tff(fact_3409_ivl__subset,axiom,
    ! [I: int,J2: int,M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or4662586982721622107an_int(I,J2)),set_or4662586982721622107an_int(M,N)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J2),I))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J2),N)) ) ) ) ).

% ivl_subset
tff(fact_3410_ivl__subset,axiom,
    ! [I: code_integer,J2: code_integer,M: code_integer,N: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I,J2)),set_or8404916559141939852nteger(M,N)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,J2),I))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,M),I))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,J2),N)) ) ) ) ).

% ivl_subset
tff(fact_3411_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),U))) ) ).

% T_vebt_buildupi_univ
tff(fact_3412_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T))) ).

% cnt_non_neg
tff(fact_3413_pos__mult__pos__ge,axiom,
    ! [X: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(N),one_one_int)),aa_int_int(times_times_int(N),X))) ) ) ).

% pos_mult_pos_ge
tff(fact_3414_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(bit0(one)) ).

% dbl_simps(3)
tff(fact_3415_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(bit0(one)) ).

% dbl_simps(3)
tff(fact_3416_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% dbl_simps(3)
tff(fact_3417_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% dbl_simps(3)
tff(fact_3418_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N))) ) ).

% two_realpow_ge_two
tff(fact_3419_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N) = zero_z3403309356797280102nteger )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3420_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3421_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3422_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_3423_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3424_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3425_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3426_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3427_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3428_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,A) = zero_z3403309356797280102nteger ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3429_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3430_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
tff(fact_3431_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3432_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3433_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3434_diff__zero,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,zero_z3403309356797280102nteger) = A ).

% diff_zero
tff(fact_3435_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_3436_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_3437_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_3438_diff__zero,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_zero
tff(fact_3439_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_3440_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3441_diff__0__right,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,zero_z3403309356797280102nteger) = A ).

% diff_0_right
tff(fact_3442_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3443_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_3444_diff__0__right,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3445_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3446_diff__self,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,A) = zero_z3403309356797280102nteger ).

% diff_self
tff(fact_3447_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_3448_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_3449_diff__self,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% diff_self
tff(fact_3450_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3451_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).

% add_diff_cancel
tff(fact_3452_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3453_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3454_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(minus_minus_rat(A,B),B) = A ).

% diff_add_cancel
tff(fact_3455_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3456_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_3457_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
tff(fact_3458_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3459_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_left
tff(fact_3460_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3461_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),A) = B ).

% add_diff_cancel_left'
tff(fact_3462_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3463_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3464_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_3465_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
tff(fact_3466_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3467_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_right
tff(fact_3468_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3469_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).

% add_diff_cancel_right'
tff(fact_3470_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3471_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3472_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : bit0(M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_3473_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != bit0(N) ).

% semiring_norm(89)
tff(fact_3474_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_3475_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_3476_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_3477_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_3478_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_3479_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_3480_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3481_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3482_diff__diff__left,axiom,
    ! [I: nat,J2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)) ).

% diff_diff_left
tff(fact_3483_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_3484_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(73)
tff(fact_3485_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(80)
tff(fact_3486_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_3487_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_3488_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_3489_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_3490_dbl__simps_I2_J,axiom,
    neg_nu8804712462038260780nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% dbl_simps(2)
tff(fact_3491_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3492_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3493_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3494_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_3495_zero__comp__diff__simps_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3496_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3497_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3498_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3499_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3500_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3501_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3502_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_3503_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3504_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3505_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3506_zero__comp__diff__simps_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3507_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3508_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_3509_diff__numeral__special_I9_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_3510_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_3511_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_3512_diff__numeral__special_I9_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_3513_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( plus_plus_rat(minus_minus_rat(A,B),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3514_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3515_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3516_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3517_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3518_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3519_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3520_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3521_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3522_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3523_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3524_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3525_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_3526_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_3527_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_3528_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_3529_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3530_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3531_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3532_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3533_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3534_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3535_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3536_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3537_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3538_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3539_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3540_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3541_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_3542_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% semiring_1_class.of_nat_0
tff(fact_3543_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_3544_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_3545_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_3546_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% semiring_1_class.of_nat_0
tff(fact_3547_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_rat(N) ).

% of_nat_numeral
tff(fact_3548_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_real(N) ).

% of_nat_numeral
tff(fact_3549_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_3550_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_3551_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).

% of_nat_numeral
tff(fact_3552_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_3553_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_3554_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_3555_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_3556_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_iff
tff(fact_3557_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_3558_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_3559_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_3560_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% of_nat_le_iff
tff(fact_3561_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_3562_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_3563_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_3564_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_3565_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_add
tff(fact_3566_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3567_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3568_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3569_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3570_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3571_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3572_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3573_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3574_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3575_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3576_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3577_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3578_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3579_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3580_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_3581_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% zero_less_diff
tff(fact_3582_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_3583_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3584_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_3585_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_3586_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_3587_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_3588_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_3589_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% of_nat_mult
tff(fact_3590_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_3591_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3592_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3593_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,X) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3594_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,X) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3595_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3596_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,X) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3597_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,X) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3598_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,X) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3599_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3600_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3601_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3602_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3603_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3604_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3605_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3606_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3607_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3608_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3609_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3610_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3611_of__bool__not__iff,axiom,
    ! [P: bool] : aa_bool_real(zero_n3304061248610475627l_real,aa_bool_bool(fNot,P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_bool_real(zero_n3304061248610475627l_real,P)) ).

% of_bool_not_iff
tff(fact_3612_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat(P)) ).

% of_bool_not_iff
tff(fact_3613_of__bool__not__iff,axiom,
    ! [P: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P)) = aa_int_int(minus_minus_int(one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P)) ).

% of_bool_not_iff
tff(fact_3614_of__bool__not__iff,axiom,
    ! [P: bool] : zero_n356916108424825756nteger(aa_bool_bool(fNot,P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger(P)) ).

% of_bool_not_iff
tff(fact_3615_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),bit0(N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(9)
tff(fact_3616_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bit0(M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(7)
tff(fact_3617_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),bit0(N)) = bit0(aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_3618_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,bit0(M)),aa_num_num(bit1,N)) = bit0(aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_3619_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,bit0(M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(72)
tff(fact_3620_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),bit0(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(81)
tff(fact_3621_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_3622_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N))) ).

% semiring_norm(77)
tff(fact_3623_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_real(semiri5074537144036343181t_real,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = aa_bool_real(zero_n3304061248610475627l_real,P) ).

% of_nat_of_bool
tff(fact_3624_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_complex(semiri8010041392384452111omplex,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = zero_n1201886186963655149omplex(P) ).

% of_nat_of_bool
tff(fact_3625_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_nat(semiri1316708129612266289at_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P) ).

% of_nat_of_bool
tff(fact_3626_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_int(semiri1314217659103216013at_int,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_nat_of_bool
tff(fact_3627_of__nat__of__bool,axiom,
    ! [P: bool] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_bool_nat(zero_n2687167440665602831ol_nat,P)) = zero_n356916108424825756nteger(P) ).

% of_nat_of_bool
tff(fact_3628_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(bit0(K)) ).

% dbl_simps(5)
tff(fact_3629_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(bit0(K)) ).

% dbl_simps(5)
tff(fact_3630_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bit0(K)) ).

% dbl_simps(5)
tff(fact_3631_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,bit0(K)) ).

% dbl_simps(5)
tff(fact_3632_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3633_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3634_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3635_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3636_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3637_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = plus_plus_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_3638_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_3639_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(plus_plus_int(one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_3640_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_3641_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_3642_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_3643_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_3644_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3645_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_3646_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_3647_semiring__norm_I3_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bit0(N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_3648_semiring__norm_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N)) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% semiring_norm(4)
tff(fact_3649_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bit0(M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_3650_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_3651_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),one)) ).

% semiring_norm(10)
tff(fact_3652_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),bit0(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)))) ).

% semiring_norm(16)
tff(fact_3653_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,bit0(M)),aa_num_num(bit1,N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).

% semiring_norm(79)
tff(fact_3654_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),bit0(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% semiring_norm(74)
tff(fact_3655_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3656_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3657_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3658_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3659_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% of_nat_0_less_iff
tff(fact_3660_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_3661_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3662_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3663_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3664_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3665_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3666_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3667_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3668_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3669_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3670_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3671_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3672_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3673_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3674_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3675_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3676_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3677_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3678_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3679_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3680_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3681_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3682_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3683_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3684_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3685_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3686_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3687_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3688_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3689_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3690_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3691_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3692_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3693_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),numeral_numeral_real(W)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_3694_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_3695_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_3696_even__diff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(minus_minus_int(A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B))) ) ).

% even_diff
tff(fact_3697_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3698_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3699_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3700_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3701_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3702_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_3703_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_3704_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_3705_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_3706_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3707_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3708_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3709_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3710_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3711_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3712_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3713_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3714_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3715_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3716_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3717_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3718_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3719_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3720_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3721_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3722_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3723_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3724_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3725_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3726_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ).

% even_of_nat
tff(fact_3727_even__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ) ).

% even_of_nat
tff(fact_3728_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_3729_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).

% even_diff_nat
tff(fact_3730_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N),one_one_Code_integer)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3731_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3732_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),one_one_int)))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3733_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_3734_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_3735_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_3736_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X3: real] : pp(aa_set_real_bool(member_real(X3),S2))
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),S2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Z4)) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( pp(aa_set_real_bool(member_real(X3),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Y3)) )
            & ! [Z4: real] :
                ( ! [X2: real] :
                    ( pp(aa_set_real_bool(member_real(X2),S2))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Z4)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z4)) ) ) ) ) ).

% complete_real
tff(fact_3737_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_3738_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_3739_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_3740_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_3741_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ) ) ).

% of_nat_diff
tff(fact_3742_le__refl,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).

% le_refl
tff(fact_3743_le__trans,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_3744_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% eq_imp_le
tff(fact_3745_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_3746_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_3747_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_3748_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3749_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).

% diff_le_mono
tff(fact_3750_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

% diff_le_self
tff(fact_3751_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_3752_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_3753_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).

% nat_le_linear
tff(fact_3754_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => ? [X2: nat] :
            ( pp(aa_nat_bool(P,X2))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X2)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_3755_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3756_cancel__ab__semigroup__add__class_Odiff__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) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3757_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3758_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3759_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3760_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
tff(fact_3761_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3762_diff__commute,axiom,
    ! [I: nat,J2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J2) ).

% diff_commute
tff(fact_3763_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J2))) ) ).

% of_nat_mono
tff(fact_3764_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J2))) ) ).

% of_nat_mono
tff(fact_3765_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J2))) ) ).

% of_nat_mono
tff(fact_3766_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_3767_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_3768_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_3769_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_3770_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_3771_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_3772_eq__iff__diff__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( minus_8373710615458151222nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_iff_diff_eq_0
tff(fact_3773_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3774_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3775_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(minus_minus_int(A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3776_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3777_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3778_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3779_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C))) ) ).

% diff_right_mono
tff(fact_3780_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_right_mono
tff(fact_3781_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C))) ) ).

% diff_right_mono
tff(fact_3782_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B))) ) ).

% diff_left_mono
tff(fact_3783_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_left_mono
tff(fact_3784_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B))) ) ).

% diff_left_mono
tff(fact_3785_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D))) ) ) ).

% diff_mono
tff(fact_3786_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_mono
tff(fact_3787_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D))) ) ) ).

% diff_mono
tff(fact_3788_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_strict_right_mono
tff(fact_3789_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C))) ) ).

% diff_strict_right_mono
tff(fact_3790_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C))) ) ).

% diff_strict_right_mono
tff(fact_3791_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,C))) ) ).

% diff_strict_right_mono
tff(fact_3792_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_strict_left_mono
tff(fact_3793_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B))) ) ).

% diff_strict_left_mono
tff(fact_3794_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B))) ) ).

% diff_strict_left_mono
tff(fact_3795_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(C,A)),minus_8373710615458151222nteger(C,B))) ) ).

% diff_strict_left_mono
tff(fact_3796_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3797_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3798_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3799_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( minus_8373710615458151222nteger(A,B) = minus_8373710615458151222nteger(C,D) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_3800_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_strict_mono
tff(fact_3801_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D))) ) ) ).

% diff_strict_mono
tff(fact_3802_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D))) ) ) ).

% diff_strict_mono
tff(fact_3803_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,D),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,D))) ) ) ).

% diff_strict_mono
tff(fact_3804_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_3805_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
tff(fact_3806_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) = aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),aa_int_int(minus_minus_int(C),D)) ).

% add_diff_add
tff(fact_3807_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3808_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( minus_minus_rat(A2,B) = plus_plus_rat(K,minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_3809_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3810_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_3811_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
tff(fact_3812_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = C )
    <=> ( A = aa_int_int(plus_plus_int(C),B) ) ) ).

% diff_eq_eq
tff(fact_3813_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3814_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
tff(fact_3815_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(minus_minus_int(C),B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3816_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_3817_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
tff(fact_3818_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),C) ).

% add_diff_eq
tff(fact_3819_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_3820_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
tff(fact_3821_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_diff_eq2
tff(fact_3822_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_3823_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
tff(fact_3824_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_add_eq
tff(fact_3825_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3826_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
tff(fact_3827_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3828_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_3829_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
tff(fact_3830_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_3831_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(C),B) = A )
     => ( C = aa_int_int(minus_minus_int(A),B) ) ) ).

% add_implies_diff
tff(fact_3832_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_3833_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
tff(fact_3834_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_3835_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% diff_diff_eq
tff(fact_3836_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D4: real,Q: fun_real_bool] :
      ( ! [X2: real,K2: real] :
          ( pp(aa_real_bool(P,X2))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
     => ( ! [X2: real,K2: real] :
            ( pp(aa_real_bool(Q,X2))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
       => ! [X3: real,K5: real] :
            ( ( pp(aa_real_bool(P,X3))
              & pp(aa_real_bool(Q,X3)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D4))))
              & pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3837_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D4: rat,Q: fun_rat_bool] :
      ( ! [X2: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X2))
        <=> pp(aa_rat_bool(P,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K2),D4)))) )
     => ( ! [X2: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X2))
          <=> pp(aa_rat_bool(Q,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K2),D4)))) )
       => ! [X3: rat,K5: rat] :
            ( ( pp(aa_rat_bool(P,X3))
              & pp(aa_rat_bool(Q,X3)) )
          <=> ( pp(aa_rat_bool(P,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D4))))
              & pp(aa_rat_bool(Q,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3838_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D4: int,Q: fun_int_bool] :
      ( ! [X2: int,K2: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D4)))) )
     => ( ! [X2: int,K2: int] :
            ( pp(aa_int_bool(Q,X2))
          <=> pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D4)))) )
       => ! [X3: int,K5: int] :
            ( ( pp(aa_int_bool(P,X3))
              & pp(aa_int_bool(Q,X3)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K5),D4))))
              & pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K5),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_3839_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D4: real,Q: fun_real_bool] :
      ( ! [X2: real,K2: real] :
          ( pp(aa_real_bool(P,X2))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
     => ( ! [X2: real,K2: real] :
            ( pp(aa_real_bool(Q,X2))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
       => ! [X3: real,K5: real] :
            ( ( pp(aa_real_bool(P,X3))
              | pp(aa_real_bool(Q,X3)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D4))))
              | pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3840_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D4: rat,Q: fun_rat_bool] :
      ( ! [X2: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X2))
        <=> pp(aa_rat_bool(P,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K2),D4)))) )
     => ( ! [X2: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X2))
          <=> pp(aa_rat_bool(Q,minus_minus_rat(X2,aa_rat_rat(times_times_rat(K2),D4)))) )
       => ! [X3: rat,K5: rat] :
            ( ( pp(aa_rat_bool(P,X3))
              | pp(aa_rat_bool(Q,X3)) )
          <=> ( pp(aa_rat_bool(P,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D4))))
              | pp(aa_rat_bool(Q,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3841_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D4: int,Q: fun_int_bool] :
      ( ! [X2: int,K2: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D4)))) )
     => ( ! [X2: int,K2: int] :
            ( pp(aa_int_bool(Q,X2))
          <=> pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D4)))) )
       => ! [X3: int,K5: int] :
            ( ( pp(aa_int_bool(P,X3))
              | pp(aa_int_bool(Q,X3)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K5),D4))))
              | pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K5),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_3842_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_3843_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_3844_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_3845_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib'
tff(fact_3846_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_3847_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(times_times_rat(B),A),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_3848_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_3849_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),C)),A) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A)) ).

% left_diff_distrib'
tff(fact_3850_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_3851_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_3852_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib
tff(fact_3853_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_3854_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_3855_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% left_diff_distrib
tff(fact_3856_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_3857_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_3858_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_3859_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
tff(fact_3860_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_3861_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : bit0(X22) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_3862_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_3863_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Z))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))) ) ) ).

% dvd_diff
tff(fact_3864_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Z))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),minus_minus_rat(Y,Z))) ) ) ).

% dvd_diff
tff(fact_3865_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(minus_minus_int(Y),Z))) ) ) ).

% dvd_diff
tff(fact_3866_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(minus_minus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(minus_minus_int(B),C))) ) ).

% dvd_diff_commute
tff(fact_3867_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
           => pp(aa_nat_bool(P,N2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_3868_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_3869_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_3870_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_right_eq
tff(fact_3871_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_3872_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_left_eq
tff(fact_3873_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_3874_mod__diff__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A3),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_3875_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B3),C) ) ) ) ).

% mod_diff_cong
tff(fact_3876_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_eq
tff(fact_3877_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_3878_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_3879_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_3880_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N)) ) ) ).

% diff_diff_less
tff(fact_3881_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),N)),K)) ) ).

% less_imp_diff_less
tff(fact_3882_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_3883_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_3884_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).

% diff_less_mono
tff(fact_3885_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_3886_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

% diff_add_inverse
tff(fact_3887_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_3888_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_3889_le__diff__conv,axiom,
    ! [J2: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_3890_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2)) ) ) ).

% Nat.le_diff_conv2
tff(fact_3891_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3892_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3893_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I) = K )
      <=> ( J2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3894_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% diff_mult_distrib2
tff(fact_3895_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% diff_mult_distrib
tff(fact_3896_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_3897_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_3898_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_3899_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_3900_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_3901_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% field_char_0_class.of_nat_div
tff(fact_3902_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% field_char_0_class.of_nat_div
tff(fact_3903_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_complex(semiri8010041392384452111omplex,N)) ).

% field_char_0_class.of_nat_div
tff(fact_3904_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% of_nat_0_le_iff
tff(fact_3905_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ).

% of_nat_0_le_iff
tff(fact_3906_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% of_nat_0_le_iff
tff(fact_3907_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).

% of_nat_0_le_iff
tff(fact_3908_of__nat__0__le__iff,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% of_nat_0_le_iff
tff(fact_3909_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_3910_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)) ).

% of_nat_less_0_iff
tff(fact_3911_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_3912_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_3913_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_3914_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3915_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N)) != zero_z3403309356797280102nteger ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3916_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3917_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3918_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3919_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3920_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_3921_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_3922_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_3923_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_3924_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% of_nat_less_imp_less
tff(fact_3925_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).

% less_imp_of_nat_less
tff(fact_3926_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) ) ).

% less_imp_of_nat_less
tff(fact_3927_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).

% less_imp_of_nat_less
tff(fact_3928_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% less_imp_of_nat_less
tff(fact_3929_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).

% less_imp_of_nat_less
tff(fact_3930_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_3931_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% div_mult2_eq'
tff(fact_3932_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,A),aa_nat_rat(semiri681578069525770553at_rat,B)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_3933_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A)),aa_nat_real(semiri5074537144036343181t_real,B)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_3934_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_3935_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A)),aa_nat_nat(semiri1316708129612266289at_nat,B)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_3936_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,A)),aa_nat_complex(semiri8010041392384452111omplex,B)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_3937_le__iff__diff__le__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,B)),zero_z3403309356797280102nteger)) ) ).

% le_iff_diff_le_0
tff(fact_3938_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_3939_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_3940_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(A),B)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_3941_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_3942_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_3943_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(A),B)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_3944_less__iff__diff__less__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,B)),zero_z3403309356797280102nteger)) ) ).

% less_iff_diff_less_0
tff(fact_3945_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3946_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3947_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3948_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3949_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3950_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3951_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3952_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3953_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).

% le_add_diff
tff(fact_3954_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_3955_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),minus_minus_rat(C,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),C)) ) ).

% le_diff_eq
tff(fact_3956_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% le_diff_eq
tff(fact_3957_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(minus_minus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),C)) ) ).

% le_diff_eq
tff(fact_3958_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(C,B))) ) ).

% diff_le_eq
tff(fact_3959_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_le_eq
tff(fact_3960_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(plus_plus_int(C),B))) ) ).

% diff_le_eq
tff(fact_3961_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),minus_minus_rat(N,K))) ) ).

% add_le_imp_le_diff
tff(fact_3962_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3963_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3964_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(minus_minus_int(N),K))) ) ).

% add_le_imp_le_diff
tff(fact_3965_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),plus_plus_rat(J2,K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),plus_plus_rat(J2,K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(N,K)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3966_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3967_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3968_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(plus_plus_int(J2),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(plus_plus_int(J2),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(N),K)),J2)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3969_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3970_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3971_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3972_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3973_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( plus_p5714425477246183910nteger(B,minus_8373710615458151222nteger(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3974_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% less_diff_eq
tff(fact_3975_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),minus_minus_rat(C,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),C)) ) ).

% less_diff_eq
tff(fact_3976_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(minus_minus_int(C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(A),B)),C)) ) ).

% less_diff_eq
tff(fact_3977_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),minus_8373710615458151222nteger(C,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(A,B)),C)) ) ).

% less_diff_eq
tff(fact_3978_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_less_eq
tff(fact_3979_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(C,B))) ) ).

% diff_less_eq
tff(fact_3980_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(plus_plus_int(C),B))) ) ).

% diff_less_eq
tff(fact_3981_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,B)),C))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),plus_p5714425477246183910nteger(C,B))) ) ).

% diff_less_eq
tff(fact_3982_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A)),B)) ).

% mult_diff_mult
tff(fact_3983_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),Y),aa_rat_rat(times_times_rat(A),B)) = plus_plus_rat(aa_rat_rat(times_times_rat(X),minus_minus_rat(Y,B)),aa_rat_rat(times_times_rat(minus_minus_rat(X,A)),B)) ).

% mult_diff_mult
tff(fact_3984_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(X),Y)),aa_int_int(times_times_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(X),aa_int_int(minus_minus_int(Y),B))),aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),A)),B)) ).

% mult_diff_mult
tff(fact_3985_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% square_diff_square_factored
tff(fact_3986_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(plus_plus_rat(X,Y)),minus_minus_rat(X,Y)) ).

% square_diff_square_factored
tff(fact_3987_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(X),X)),aa_int_int(times_times_int(Y),Y)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(X),Y)),aa_int_int(minus_minus_int(X),Y)) ).

% square_diff_square_factored
tff(fact_3988_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_3989_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
    <=> ( C = plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D) ) ) ).

% eq_add_iff2
tff(fact_3990_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D) )
    <=> ( C = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_3991_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3992_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C) = D ) ) ).

% eq_add_iff1
tff(fact_3993_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3994_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3995_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3996_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != bit0(X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_3997_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : X != product_Pair_num_num(one,bit0(N2))
       => ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M3: num] : X != product_Pair_num_num(bit0(M3),one)
           => ( ! [M3: num,N2: num] : X != product_Pair_num_num(bit0(M3),bit0(N2))
             => ( ! [M3: num,N2: num] : X != product_Pair_num_num(bit0(M3),aa_num_num(bit1,N2))
               => ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),one)
                 => ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),bit0(N2))
                   => ~ ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_3998_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_3999_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% of_nat_dvd_iff
tff(fact_4000_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))) ).

% dvd_minus_mod
tff(fact_4001_dvd__minus__mod,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)))) ).

% dvd_minus_mod
tff(fact_4002_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)))) ).

% dvd_minus_mod
tff(fact_4003_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(minus_minus_int(A),B))) ) ).

% mod_eq_dvd_iff
tff(fact_4004_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B))) ) ).

% mod_eq_dvd_iff
tff(fact_4005_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mod
tff(fact_4006_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mod
tff(fact_4007_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mod
tff(fact_4008_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(suc,N) = M )
     => ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_4009_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_4010_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_4011_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_4012_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_4013_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X)))) ).

% real_of_nat_div4
tff(fact_4014_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_4015_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_4016_less__diff__conv,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2)) ) ).

% less_diff_conv
tff(fact_4017_less__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_4018_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% dbl_def
tff(fact_4019_dbl__def,axiom,
    ! [X: rat] : neg_numeral_dbl_rat(X) = plus_plus_rat(X,X) ).

% dbl_def
tff(fact_4020_dbl__def,axiom,
    ! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(plus_plus_int(X),X) ).

% dbl_def
tff(fact_4021_nat__eq__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_4022_nat__eq__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4023_nat__le__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_4024_nat__le__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_4025_nat__diff__add__eq1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_4026_nat__diff__add__eq2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_4027_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_4028_nat__minus__mod,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_4029_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_4030_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

% mod_if
tff(fact_4031_mod__nat__sub,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y) ) ) ).

% mod_nat_sub
tff(fact_4032_nat__minus__mod__plus__right,axiom,
    ! [N: nat,X: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) ).

% nat_minus_mod_plus_right
tff(fact_4033_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X2: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D3),B))
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D3 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_4034_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4035_small__powers__of__2,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),one_one_nat)))) ) ).

% small_powers_of_2
tff(fact_4036_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4037_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,K) != zero_z3403309356797280102nteger )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4038_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4039_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4040_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4041_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,K) != zero_zero_complex )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K)) ) ).

% of_nat_gt_0
tff(fact_4042_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4043_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4044_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4045_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4046_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4047_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4048_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D))) ) ).

% less_add_iff2
tff(fact_4049_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D))) ) ).

% less_add_iff2
tff(fact_4050_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D))) ) ).

% less_add_iff2
tff(fact_4051_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(B,A)),E),D))) ) ).

% less_add_iff2
tff(fact_4052_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D)) ) ).

% less_add_iff1
tff(fact_4053_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D)) ) ).

% less_add_iff1
tff(fact_4054_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C)),D)) ) ).

% less_add_iff1
tff(fact_4055_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B)),E),C)),D)) ) ).

% less_add_iff1
tff(fact_4056_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4057_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4058_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(minus_minus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4059_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4060_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4061_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4062_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4063_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4064_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4065_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4066_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4067_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( minus_minus_rat(A,divide_divide_rat(B,Z)) = A ) )
      & ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(A,divide_divide_rat(B,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4068_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).

% square_diff_one_factored
tff(fact_4069_square__diff__one__factored,axiom,
    ! [X: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),one_one_rat) = aa_rat_rat(times_times_rat(plus_plus_rat(X,one_one_rat)),minus_minus_rat(X,one_one_rat)) ).

% square_diff_one_factored
tff(fact_4070_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(X),X)),one_one_int) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(X),one_one_int)),aa_int_int(minus_minus_int(X),one_one_int)) ).

% square_diff_one_factored
tff(fact_4071_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D4))
     => ! [X3: real,K5: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D4))),T))) ) ) ).

% inf_period(4)
tff(fact_4072_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D4))
     => ! [X3: rat,K5: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D4)),T))) ) ) ).

% inf_period(4)
tff(fact_4073_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X3: int,K5: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K5),D4))),T))) ) ) ).

% inf_period(4)
tff(fact_4074_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D4))
     => ! [X3: real,K5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D4))),T))) ) ) ).

% inf_period(3)
tff(fact_4075_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D4))
     => ! [X3: rat,K5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X3,T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X3,aa_rat_rat(times_times_rat(K5),D4)),T))) ) ) ).

% inf_period(3)
tff(fact_4076_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X3: int,K5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K5),D4))),T))) ) ) ).

% inf_period(3)
tff(fact_4077_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4078_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_4079_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4080_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4081_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4082_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4083_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4084_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4085_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4086_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4087_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_4088_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4089_zmde,axiom,
    ! [B: int,A: int] : aa_int_int(times_times_int(B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) = aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) ).

% zmde
tff(fact_4090_zmde,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)) = minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) ).

% zmde
tff(fact_4091_numeral__Bit1,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_Bit1
tff(fact_4092_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_Bit1
tff(fact_4093_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).

% numeral_Bit1
tff(fact_4094_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_4095_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_4096_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bit0(N))) ).

% eval_nat_numeral(3)
tff(fact_4097_power__diff,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% power_diff
tff(fact_4098_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% power_diff
tff(fact_4099_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).

% power_diff
tff(fact_4100_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% power_diff
tff(fact_4101_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_diff
tff(fact_4102_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_diff
tff(fact_4103_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(M))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4104_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4105_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(M)),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4106_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,bit0(N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4107_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(N))),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4108_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(N)),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4109_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4110_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4111_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4112_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y4: real] :
        ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% reals_Archimedean3
tff(fact_4113_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_4114_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4115_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4116_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M)) ) ) ).

% nz_le_conv_less
tff(fact_4117_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% add_eq_if
tff(fact_4118_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_4119_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_4120_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),N))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_4121_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

% mult_eq_if
tff(fact_4122_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_4123_nat__less__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_4124_nat__less__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_4125_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z) ) ) ) ) ) ).

% mod_nat_add
tff(fact_4126_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_4127_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),D))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D),B))) ) ) ).

% diff_mod_le
tff(fact_4128_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4129_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)) ).

% modulo_nat_def
tff(fact_4130_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc8114805097830739791at_num] :
      ~ ! [F3: fun_nat_fun_num_num,A4: nat,B4: nat,Acc: num] : X != produc460779904311030791at_num(F3,produc1195630363706982562at_num(A4,product_Pair_nat_num(B4,Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4131_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc9217583074051984497at_nat] :
      ~ ! [F3: fun_nat_fun_nat_nat,A4: nat,B4: nat,Acc: nat] : X != produc2818902965508115497at_nat(F3,produc487386426758144856at_nat(A4,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B4),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4132_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_4133_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( ( N != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).

% inverse_of_nat_le
tff(fact_4134_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_4135_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_4136_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_4137_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_4138_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_4139_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_4140_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_4141_power2__commute,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),X)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_commute
tff(fact_4142_power2__commute,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(X,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Y,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_commute
tff(fact_4143_power2__commute,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_commute
tff(fact_4144_power2__commute,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_commute
tff(fact_4145_power2__commute,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Y),X)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_commute
tff(fact_4146_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% odd_mod_4_div_2
tff(fact_4147_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit1_div_2
tff(fact_4148_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,bit0(one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit1_div_2
tff(fact_4149_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M3)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_4150_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_4151_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,bit0(Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_4152_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(bit0(Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_4153_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_4154_odd__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) ).

% odd_numeral
tff(fact_4155_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4156_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4157_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4158_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_4159_power3__eq__cube,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A)),A) ).

% power3_eq_cube
tff(fact_4160_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).

% power3_eq_cube
tff(fact_4161_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).

% power3_eq_cube
tff(fact_4162_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_4163_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(times_times_int(aa_int_int(times_times_int(A),A)),A) ).

% power3_eq_cube
tff(fact_4164_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_4165_power__eq__if,axiom,
    ! [M: nat,P3: assn] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_assn(power_power_assn(P3),M) = one_one_assn ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_assn(power_power_assn(P3),M) = aa_assn_assn(times_times_assn(P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4166_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4167_power__eq__if,axiom,
    ! [M: nat,P3: code_integer] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = one_one_Code_integer ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(P3),aa_nat_Code_integer(power_8256067586552552935nteger(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4168_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4169_power__eq__if,axiom,
    ! [M: nat,P3: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = aa_rat_rat(times_times_rat(P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4170_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4171_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = aa_int_int(times_times_int(P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_4172_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_4173_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% power_minus_mult
tff(fact_4174_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% power_minus_mult
tff(fact_4175_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% power_minus_mult
tff(fact_4176_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_minus_mult
tff(fact_4177_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_4178_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_4179_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).

% diff_le_diff_pow
tff(fact_4180_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_4181_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_if
tff(fact_4182_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_4183_power__sub,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_sub
tff(fact_4184_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))) ) ) ).

% nat_mult_power_less_eq
tff(fact_4185_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N))) ).

% of_nat_less_two_power
tff(fact_4186_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(numeral_numeral_rat(bit0(one))),N))) ).

% of_nat_less_two_power
tff(fact_4187_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N))) ).

% of_nat_less_two_power
tff(fact_4188_of__nat__less__two__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ).

% of_nat_less_two_power
tff(fact_4189_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_4190_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_4191_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(U,divide_divide_rat(aa_rat_rat(times_times_rat(R2),minus_minus_rat(V,U)),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4192_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4193_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4194_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4195_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4196_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4197_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4198_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4199_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4200_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4201_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4202_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_4203_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) ) ) ).

% power_minus_is_div
tff(fact_4204_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ) ).

% nat_le_power_trans
tff(fact_4205_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_4206_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_4207_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4208_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4209_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4210_less__two__pow__divI,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))) ) ) ).

% less_two_pow_divI
tff(fact_4211_less__two__pow__divD,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% less_two_pow_divD
tff(fact_4212_nat__power__less__diff,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))) ) ).

% nat_power_less_diff
tff(fact_4213_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))) ) ) ).

% nat_less_power_trans
tff(fact_4214_power__mod__div,axiom,
    ! [X: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ).

% power_mod_div
tff(fact_4215_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(X,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_4216_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_4217_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_4218_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = minus_minus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_4219_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(X),Y)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_4220_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_4221_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_4222_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% even_mask_div_iff'
tff(fact_4223_atLeastLessThan__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or66887138388493659n_real(A,B)),set_or66887138388493659n_real(C,D)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4224_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or1222409239386451017an_num(A,B)),set_or1222409239386451017an_num(C,D)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4225_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(C),D)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4226_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or4662586982721622107an_int(A,B)),set_or4662586982721622107an_int(C,D)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4227_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A,B)),set_or8404916559141939852nteger(C,D)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),D)) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4228_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4229_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4230_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4231_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4232_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4233_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
       => ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4234_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4235_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4236_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4237_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C),D) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4238_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4239_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4240_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4241_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4242_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4243_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C),D) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4244_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4245_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D))
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4246_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4247_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4248_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4249_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))))
         => ( aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4250_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N))))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N) = zero_z3403309356797280102nteger )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_4251_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_4252_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).

% even_mask_div_iff
tff(fact_4253_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),zero_zero_nat))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4254_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4255_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] : divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),zero_z3403309356797280102nteger))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).

% exp_div_exp_eq
tff(fact_4256_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% even_mod_4_div_2
tff(fact_4257_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_4258_space__cnt,axiom,
    ! [T: vEBT_VEBT] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)))) ).

% space_cnt
tff(fact_4259_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))),U))) ) ).

% vebt_buildup_bound
tff(fact_4260_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% Tb_T_vebt_buildupi''
tff(fact_4261_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))))) ).

% T_vebt_buildupi_cnt'
tff(fact_4262_count__buildup_H,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))) ).

% count_buildup'
tff(fact_4263_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Q2)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_4264_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Q2)),one_one_int),aa_int_int(minus_minus_int(R2),aa_num_int(numeral_numeral_int,L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Q2),R2) ) ) ) ).

% divmod_step_eq
tff(fact_4265_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2))
       => ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Q2),R2) ) ) ) ).

% divmod_step_eq
tff(fact_4266_count__buildup,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N)))) ).

% count_buildup
tff(fact_4267_idiff__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_4268_idiff__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4269_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_4270_ivl__diff,axiom,
    ! [I: real,N: real,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),N))
     => ( minus_minus_set_real(set_or66887138388493659n_real(I,M),set_or66887138388493659n_real(I,N)) = set_or66887138388493659n_real(N,M) ) ) ).

% ivl_diff
tff(fact_4271_ivl__diff,axiom,
    ! [I: num,N: num,M: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),N))
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I,M),set_or1222409239386451017an_num(I,N)) = set_or1222409239386451017an_num(N,M) ) ) ).

% ivl_diff
tff(fact_4272_ivl__diff,axiom,
    ! [I: nat,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
     => ( minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),M),aa_nat_set_nat(set_or4665077453230672383an_nat(I),N)) = aa_nat_set_nat(set_or4665077453230672383an_nat(N),M) ) ) ).

% ivl_diff
tff(fact_4273_ivl__diff,axiom,
    ! [I: int,N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),N))
     => ( minus_minus_set_int(set_or4662586982721622107an_int(I,M),set_or4662586982721622107an_int(I,N)) = set_or4662586982721622107an_int(N,M) ) ) ).

% ivl_diff
tff(fact_4274_ivl__diff,axiom,
    ! [I: code_integer,N: code_integer,M: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),N))
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I,M),set_or8404916559141939852nteger(I,N)) = set_or8404916559141939852nteger(N,M) ) ) ).

% ivl_diff
tff(fact_4275_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).

% int_dvd_int_iff
tff(fact_4276_Tb_H__cnt,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_vebt_buildup(N))))) ).

% Tb'_cnt
tff(fact_4277_t__buildup__cnt,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),numeral_numeral_real(aa_num_num(bit1,bit0(aa_num_num(bit1,one))))))) ).

% t_buildup_cnt
tff(fact_4278_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),aa_int_int(minus_minus_int(Z),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% zle_diff1_eq
tff(fact_4279_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(minus_minus_int(K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_4280_int__distrib_I3_J,axiom,
    ! [Z1: int,Z2: int,W: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(Z1),Z2)),W) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z2),W)) ).

% int_distrib(3)
tff(fact_4281_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z2: int] : aa_int_int(times_times_int(W),aa_int_int(minus_minus_int(Z1),Z2)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z2)) ).

% int_distrib(4)
tff(fact_4282_Word_Omod__minus__cong,axiom,
    ! [B: int,B3: int,X: int,X6: int,Y: int,Y5: int,Z3: int] :
      ( ( B = B3 )
     => ( ( aa_int_int(modulo_modulo_int(X),B3) = aa_int_int(modulo_modulo_int(X6),B3) )
       => ( ( aa_int_int(modulo_modulo_int(Y),B3) = aa_int_int(modulo_modulo_int(Y5),B3) )
         => ( ( aa_int_int(minus_minus_int(X6),Y5) = Z3 )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(X),Y)),B) = aa_int_int(modulo_modulo_int(Z3),B3) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_4283_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(minus_minus_int(aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(minus_minus_int(K),L)) ).

% signed_take_bit_diff
tff(fact_4284_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(minus_minus_int(M),N)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_4285_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).

% int_ops(6)
tff(fact_4286_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_4287_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_4288_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_4289_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(2)
tff(fact_4290_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).

% zle_int
tff(fact_4291_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).

% zadd_int_left
tff(fact_4292_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_4293_int__plus,axiom,
    ! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_4294_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N2: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% nonneg_int_cases
tff(fact_4295_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N2: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ).

% zero_le_imp_eq_int
tff(fact_4296_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_4297_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z))
    <=> ? [N7: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,N7)) ) ).

% zle_iff_zadd
tff(fact_4298_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_4299_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(K),one_one_int)))
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_4300_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_4301_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_4302_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y))
     => ( plus_p3455044024723400733d_enat(X,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_4303_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_4304_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_4305_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
    <=> ? [N7: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N7))) ) ).

% zless_iff_Suc_zadd
tff(fact_4306_plusinfinity,axiom,
    ! [D: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X2: int,K2: int] :
            ( pp(aa_int_bool(P2,X2))
          <=> pp(aa_int_bool(P2,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D)))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X2))
             => ( pp(aa_int_bool(P,X2))
              <=> pp(aa_int_bool(P2,X2)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P2,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_4307_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X2: int,K2: int] :
            ( pp(aa_int_bool(P1,X2))
          <=> pp(aa_int_bool(P1,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D)))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z4))
             => ( pp(aa_int_bool(P,X2))
              <=> pp(aa_int_bool(P1,X2)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P1,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_4308_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(N),one_one_int)),M)) ) ).

% less_1_helper
tff(fact_4309_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I3))
           => ( pp(aa_int_bool(P,I3))
             => pp(aa_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int))) ) )
       => ( ! [I3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),K))
             => ( pp(aa_int_bool(P,I3))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(I3),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_4310_int__mod__le_H,axiom,
    ! [B: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(B),N)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(B),N)),aa_int_int(minus_minus_int(B),N))) ) ).

% int_mod_le'
tff(fact_4311_mod__div__equality__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) = aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_equality_div_eq
tff(fact_4312_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% pos_int_cases
tff(fact_4313_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_4314_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J2: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),J2))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_4315_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X2: int] :
            ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X3: int] :
              ( pp(aa_int_bool(P,X3))
             => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4316_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4317_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),M))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(N),one_one_int)),M) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M)),one_one_int) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N))
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(N),one_one_int)),M) = aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M) ) ) ) ) ).

% int_div_sub_1
tff(fact_4318_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(minus_minus_int(K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),L))) ) ).

% even_diff_iff
tff(fact_4319_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))))) ).

% real_of_nat_div2
tff(fact_4320_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_4321_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
             => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
                 => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(X),Y)),Z) = aa_int_int(minus_minus_int(X),Y) ) )
                & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
                 => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(X),Y)),Z) = aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X),Y)),Z) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_4322_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
             => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(X),Y)),Z))
                 => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(X),Y)),Z) = aa_int_int(plus_plus_int(X),Y) ) )
                & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(X),Y)),Z))
                 => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(X),Y)),Z) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(X),Y)),Z) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_4323_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_4324_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_4325_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4326_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_4327_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_4328_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4329_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_4330_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_4331_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_4332_space__space_H,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))) ).

% space_space'
tff(fact_4333_t__build__cnt,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),numeral_numeral_real(aa_num_num(bit1,bit0(aa_num_num(bit1,one))))))) ).

% t_build_cnt
tff(fact_4334_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(bit0(one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(bit0(one))) ).

% real_average_minus_second
tff(fact_4335_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(bit0(one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(bit0(one))) ).

% real_average_minus_first
tff(fact_4336_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_int_int(minus_minus_int(aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% Tb_T_vebt_buildupi
tff(fact_4337_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),N))) ) ).

% linear_plus_1_le_power
tff(fact_4338_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,bit0(one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_4339_buildup__build__time,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))) ).

% buildup_build_time
tff(fact_4340_Tb__Tb_H,axiom,
    ! [T: nat] : aa_nat_int(vEBT_VEBT_Tb,T) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,T)) ).

% Tb_Tb'
tff(fact_4341_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N2: nat] : Z != aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_4342_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_4343_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z) ) ) ).

% eq_diff_eq'
tff(fact_4344_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_4345_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_4346_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_4347_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_4348_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A4: real,B4: real,C2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B4),C2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),B4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),C2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),C2)) ) ) ) )
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
               => ? [D6: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D6))
                    & ! [A4: real,B4: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),X2))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B4))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A4)),D6)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B4)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_4349_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,bit0(one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,bit0(one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) ) ).

% VEBT_internal.Tb.simps(3)
tff(fact_4350_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,bit0(one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,bit0(one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_4351_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,N)),aa_int_int(minus_minus_int(aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% Tb_T_vebt_buildupi'
tff(fact_4352_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N)),one_one_real)))) ) ).

% space_2_pow_bound
tff(fact_4353_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N)) = aa_nat_int(vEBT_V9176841429113362141ildupi,N) ).

% Tbuildupi_buildupi'
tff(fact_4354_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))))))) ) ).

% cnt_bound
tff(fact_4355_nat__approx__posE,axiom,
    ! [E: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
     => ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E)) ) ).

% nat_approx_posE
tff(fact_4356_nat__approx__posE,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
     => ~ ! [N2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E)) ) ).

% nat_approx_posE
tff(fact_4357_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N)),one_one_real)))) ) ).

% cnt_bound'
tff(fact_4358_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                     => ( Y = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one)))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_4359_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_0_not
tff(fact_4360_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_4361_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% deg_not_0
tff(fact_4362_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_buildup(N)),N)) ) ).

% buildup_gives_valid
tff(fact_4363_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(aa_num_num(bit1,one))))),U))) ) ) ).

% space'_bound
tff(fact_4364_space__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(aa_num_num(bit1,one))))),U))) ) ) ).

% space_bound
tff(fact_4365_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_4366_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_4367_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(P,N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_4368_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% reals_Archimedean2
tff(fact_4369_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% reals_Archimedean2
tff(fact_4370_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_4371_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_4372_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one)))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_4373_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ).

% helpyd
tff(fact_4374_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ).

% helpypredd
tff(fact_4375_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ).

% two_powr_height_bound_deg
tff(fact_4376_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_4377_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ).

% member_bound
tff(fact_4378_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).

% set_n_deg_not_0
tff(fact_4379_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ) ).

% misiz
tff(fact_4380_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_4381_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2),S3) ) ).

% deg_SUcn_Node
tff(fact_4382_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% mint_member
tff(fact_4383_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] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_4384_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summary))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)))) ).

% height_compose_summary
tff(fact_4385_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_4386_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( member_VEBT_VEBT(T,set_VEBT_VEBT2(TreeList))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)))) ) ).

% height_compose_child
tff(fact_4387_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_4388_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_4389_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_4390_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M3: nat] :
              ( ( some_nat(M3) = vEBT_vebt_mint(Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_4391_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B2)
    <=> ! [X5: complex] :
          ( member_complex(X5,set_complex2(Xs))
         => member_complex(X5,B2) ) ) ).

% subset_code(1)
tff(fact_4392_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X5,B2) ) ) ).

% subset_code(1)
tff(fact_4393_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B2))
    <=> ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
         => pp(aa_set_real_bool(member_real(X5),B2)) ) ) ).

% subset_code(1)
tff(fact_4394_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B2))
    <=> ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X5),B2)) ) ) ).

% subset_code(1)
tff(fact_4395_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),B2))
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X5),B2)) ) ) ).

% subset_code(1)
tff(fact_4396_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_complex,Xsi2: list_complex,A2: fun_co6163879640491479806x_assn,A5: fun_co6163879640491479806x_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: complex,Xi: complex] :
              ( member_complex(X2,set_complex2(Xs3))
             => ( member_complex(Xi,set_complex2(Xsi2))
               => ( aa_complex_assn(aa_com5932308198457167295x_assn(A2,X2),Xi) = aa_complex_assn(aa_com5932308198457167295x_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L4260503343685368993omplex(A2,Xs,Xsi) = vEBT_L4260503343685368993omplex(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4397_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_co2031555705788908698T_assn,A5: fun_co2031555705788908698T_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: complex,Xi: vEBT_VEBT] :
              ( member_complex(X2,set_complex2(Xs3))
             => ( member_VEBT_VEBT(Xi,set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_com5451320767305428611T_assn(A2,X2),Xi) = aa_VEBT_VEBT_assn(aa_com5451320767305428611T_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L8524933119956041985T_VEBT(A2,Xs,Xsi) = vEBT_L8524933119956041985T_VEBT(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4398_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_real,Xsi2: list_real,A2: fun_co8000059489268409724l_assn,A5: fun_co8000059489268409724l_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: complex,Xi: real] :
              ( member_complex(X2,set_complex2(Xs3))
             => ( pp(aa_set_real_bool(member_real(Xi),set_real2(Xsi2)))
               => ( aa_real_assn(aa_com7355656519465148733l_assn(A2,X2),Xi) = aa_real_assn(aa_com7355656519465148733l_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L2479436891206192927x_real(A2,Xs,Xsi) = vEBT_L2479436891206192927x_real(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4399_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_nat,Xsi2: list_nat,A2: fun_co1748589094918083360t_assn,A5: fun_co1748589094918083360t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: complex,Xi: nat] :
              ( member_complex(X2,set_complex2(Xs3))
             => ( pp(aa_set_nat_bool(member_nat(Xi),set_nat2(Xsi2)))
               => ( aa_nat_assn(aa_com1546947087700170465t_assn(A2,X2),Xi) = aa_nat_assn(aa_com1546947087700170465t_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L137475477348087235ex_nat(A2,Xs,Xsi) = vEBT_L137475477348087235ex_nat(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4400_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_int,Xsi2: list_int,A2: fun_co6164383388265235324t_assn,A5: fun_co6164383388265235324t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: complex,Xi: int] :
              ( member_complex(X2,set_complex2(Xs3))
             => ( pp(aa_set_int_bool(member_int(Xi),set_int2(Xsi2)))
               => ( aa_int_assn(aa_com2522482552906347837t_assn(A2,X2),Xi) = aa_int_assn(aa_com2522482552906347837t_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L134985006839036959ex_int(A2,Xs,Xsi) = vEBT_L134985006839036959ex_int(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4401_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_complex,Xsi2: list_complex,A2: fun_VE496882281059548826x_assn,A5: fun_VE496882281059548826x_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: complex] :
              ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs3))
             => ( member_complex(Xi,set_complex2(Xsi2))
               => ( aa_complex_assn(aa_VEB2594926317323433291x_assn(A2,X2),Xi) = aa_complex_assn(aa_VEB2594926317323433291x_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L2162147798726695391omplex(A2,Xs,Xsi) = vEBT_L2162147798726695391omplex(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4402_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,A5: fun_VE1571388752190420990T_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs3))
             => ( member_VEBT_VEBT(Xi,set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X2),Xi) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L1279224858307276611T_VEBT(A2,Xs,Xsi) = vEBT_L1279224858307276611T_VEBT(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4403_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: fun_VE6910866152645720856l_assn,A5: fun_VE6910866152645720856l_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: real] :
              ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs3))
             => ( pp(aa_set_real_bool(member_real(Xi),set_real2(Xsi2)))
               => ( aa_real_assn(aa_VEB8575627266564790985l_assn(A2,X2),Xi) = aa_real_assn(aa_VEB8575627266564790985l_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L5781919052683127133T_real(A2,Xs,Xsi) = vEBT_L5781919052683127133T_real(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4404_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE4931415397359990460t_assn,A5: fun_VE4931415397359990460t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: nat] :
              ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs3))
             => ( pp(aa_set_nat_bool(member_nat(Xi),set_nat2(Xsi2)))
               => ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,X2),Xi) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A5,X2),Xi) ) ) )
         => ( aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A2,Xs),Xsi) = aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A5,Xs3),Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4405_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: fun_VE123837653852366616t_assn,A5: fun_VE123837653852366616t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: vEBT_VEBT,Xi: int] :
              ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs3))
             => ( pp(aa_set_int_bool(member_int(Xi),set_int2(Xsi2)))
               => ( aa_int_assn(aa_VEB392321844297299145t_assn(A2,X2),Xi) = aa_int_assn(aa_VEB392321844297299145t_assn(A5,X2),Xi) ) ) )
         => ( vEBT_L8294436054247626077BT_int(A2,Xs,Xsi) = vEBT_L8294436054247626077BT_int(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_4406_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4407_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4408_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4409_length__pos__if__in__set,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4410_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4411_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4412_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(bit0(one)))))))))) ) ).

% delete_bound_height
tff(fact_4413_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),X)),one_one_nat)) ) ).

% tdeletemimi'
tff(fact_4414_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one)))))) ) ).

% tdeletemimi
tff(fact_4415_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% mint_corr
tff(fact_4416_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).

% mint_sound
tff(fact_4417_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% delete_bound_height'
tff(fact_4418_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_4419_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_4420_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
       => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_4421_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [Z6: nat] :
            ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Z6))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z6)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z6)) ) ) ) ).

% succ_member
tff(fact_4422_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [Z6: nat] :
            ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Z6))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z6),X)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z6),Y)) ) ) ) ).

% pred_member
tff(fact_4423_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_corr
tff(fact_4424_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Px) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Px)) ) ) ).

% pred_corr
tff(fact_4425_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_correct
tff(fact_4426_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% pred_correct
tff(fact_4427_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ).

% member_inv
tff(fact_4428_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_4429_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_4430_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)))) ) ).

% height_node
tff(fact_4431_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_4432_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),X) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_4433_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(bit0(one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(bit0(one)))))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))))) ) ) ).

% delete_bound_size_univ
tff(fact_4434_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool,N: nat] :
      ( ! [X2: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
         => pp(aa_VEBT_VEBTi_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
       => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,N))) ) ) ).

% inthall
tff(fact_4435_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,N: nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,set_complex2(Xs))
         => pp(aa_complex_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
       => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),N))) ) ) ).

% inthall
tff(fact_4436_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => pp(aa_VEBT_VEBT_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% inthall
tff(fact_4437_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N: nat] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => pp(aa_real_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% inthall
tff(fact_4438_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N: nat] :
      ( ! [X2: bool] :
          ( member_bool(X2,set_bool2(Xs))
         => pp(aa_bool_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% inthall
tff(fact_4439_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% inthall
tff(fact_4440_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N: nat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => pp(aa_int_bool(P,X2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% inthall
tff(fact_4441_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Ma),X))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_4442_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y6: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y6)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_4443_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ! [Y6: nat] : X != some_nat(Y6)
    <=> ( X = none_nat ) ) ).

% not_Some_eq
tff(fact_4444_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y6: num] : X != some_num(Y6)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_4445_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y6: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y6) ) ).

% not_None_eq
tff(fact_4446_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
    <=> ? [Y6: nat] : X = some_nat(Y6) ) ).

% not_None_eq
tff(fact_4447_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y6: num] : X = some_num(Y6) ) ).

% not_None_eq
tff(fact_4448_less__eq__option__None__code,axiom,
    ! [X: option_nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,none_nat),X)) ).

% less_eq_option_None_code
tff(fact_4449_less__eq__option__None__code,axiom,
    ! [X: option_num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,none_num),X)) ).

% less_eq_option_None_code
tff(fact_4450_less__option__None,axiom,
    ! [X: option_nat] : ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,X),none_nat)) ).

% less_option_None
tff(fact_4451_less__option__None,axiom,
    ! [X: option_num] : ~ pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,X),none_num)) ).

% less_option_None
tff(fact_4452_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% delete_bound_size_univ'
tff(fact_4453_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Deg) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),Deg))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% height_double_log_univ_size
tff(fact_4454_less__eq__option__Some__None,axiom,
    ! [X: nat] : ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,some_nat(X)),none_nat)) ).

% less_eq_option_Some_None
tff(fact_4455_less__eq__option__Some__None,axiom,
    ! [X: num] : ~ pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,some_num(X)),none_num)) ).

% less_eq_option_Some_None
tff(fact_4456_less__option__None__Some__code,axiom,
    ! [X: nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),some_nat(X))) ).

% less_option_None_Some_code
tff(fact_4457_less__option__None__Some__code,axiom,
    ! [X: num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,none_num),some_num(X))) ).

% less_option_None_Some_code
tff(fact_4458_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
       => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4459_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op9100744243108201484t_bool,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
       => ( ! [A4: product_prod_nat_nat,B4: nat] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_nat(B4) )
               => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4460_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op775672374849087042m_bool,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
       => ( ! [A4: product_prod_nat_nat,B4: num] :
              ( ( X = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_num(B4) )
               => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4461_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op5698422665285572208t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
       => ( ! [A4: nat,B4: product_prod_nat_nat] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4462_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op7792301606653096457t_bool,Y: option_nat] :
      ( ( ( X = none_nat )
       => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
       => ( ! [A4: nat,B4: nat] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_nat(B4) )
               => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4463_combine__options__cases,axiom,
    ! [X: option_nat,P: fun_op8690601775248757823m_bool,Y: option_num] :
      ( ( ( X = none_nat )
       => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
       => ( ! [A4: nat,B4: num] :
              ( ( X = some_nat(A4) )
             => ( ( Y = some_num(B4) )
               => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4464_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
       => ( ! [A4: num,B4: product_prod_nat_nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) )
         => pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4465_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op1064277265382982207t_bool,Y: option_nat] :
      ( ( ( X = none_num )
       => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
     => ( ( ( Y = none_nat )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
       => ( ! [A4: num,B4: nat] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_nat(B4) )
               => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) )
         => pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4466_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op1962577433978643573m_bool,Y: option_num] :
      ( ( ( X = none_num )
       => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
     => ( ( ( Y = none_num )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
       => ( ! [A4: num,B4: num] :
              ( ( X = some_num(A4) )
             => ( ( Y = some_num(B4) )
               => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) )
         => pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4467_split__option__all,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ! [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_13))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        & ! [X5: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X5))) ) ) ).

% split_option_all
tff(fact_4468_split__option__all,axiom,
    ! [P: fun_option_nat_bool] :
      ( ! [X_13: option_nat] : pp(aa_option_nat_bool(P,X_13))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        & ! [X5: nat] : pp(aa_option_nat_bool(P,some_nat(X5))) ) ) ).

% split_option_all
tff(fact_4469_split__option__all,axiom,
    ! [P: fun_option_num_bool] :
      ( ! [X_13: option_num] : pp(aa_option_num_bool(P,X_13))
    <=> ( pp(aa_option_num_bool(P,none_num))
        & ! [X5: num] : pp(aa_option_num_bool(P,some_num(X5))) ) ) ).

% split_option_all
tff(fact_4470_split__option__ex,axiom,
    ! [P: fun_op2627489687811228755t_bool] :
      ( ? [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_13))
    <=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
        | ? [X5: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X5))) ) ) ).

% split_option_ex
tff(fact_4471_split__option__ex,axiom,
    ! [P: fun_option_nat_bool] :
      ( ? [X_13: option_nat] : pp(aa_option_nat_bool(P,X_13))
    <=> ( pp(aa_option_nat_bool(P,none_nat))
        | ? [X5: nat] : pp(aa_option_nat_bool(P,some_nat(X5))) ) ) ).

% split_option_ex
tff(fact_4472_split__option__ex,axiom,
    ! [P: fun_option_num_bool] :
      ( ? [X_13: option_num] : pp(aa_option_num_bool(P,X_13))
    <=> ( pp(aa_option_num_bool(P,none_num))
        | ? [X5: num] : pp(aa_option_num_bool(P,some_num(X5))) ) ) ).

% split_option_ex
tff(fact_4473_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_4474_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_4475_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_4476_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_4477_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_4478_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_4479_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_4480_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_4481_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_4482_less__eq__option__None,axiom,
    ! [X: option_nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,none_nat),X)) ).

% less_eq_option_None
tff(fact_4483_less__eq__option__None,axiom,
    ! [X: option_num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,none_num),X)) ).

% less_eq_option_None
tff(fact_4484_less__eq__option__None__is__None,axiom,
    ! [X: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_le5914376470875661696on_nat,X),none_nat))
     => ( X = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_4485_less__eq__option__None__is__None,axiom,
    ! [X: option_num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_le6622620407824499402on_num,X),none_num))
     => ( X = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_4486_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4487_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
           => ( nth_VEBT_VEBTi(Xs,I2) = nth_VEBT_VEBTi(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4488_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
           => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4489_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Ys),I2)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4490_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4491_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4492_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I2),X_13)) )
    <=> ? [Xs4: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs4) = K )
          & ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I2),nth_VEBT_VEBT(Xs4,I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_4493_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na307722919519257988i_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
         => ? [X_13: vEBT_VEBTi] : pp(aa_VEBT_VEBTi_bool(aa_nat6176657665065580909i_bool(P,I2),X_13)) )
    <=> ? [Xs4: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs4) = K )
          & ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => pp(aa_VEBT_VEBTi_bool(aa_nat6176657665065580909i_bool(P,I2),nth_VEBT_VEBTi(Xs4,I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_4494_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6801867027684862029l_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
         => ? [X_13: real] : pp(aa_real_bool(aa_nat_fun_real_bool(P,I2),X_13)) )
    <=> ? [Xs4: list_real] :
          ( ( size_size_list_real(Xs4) = K )
          & ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => pp(aa_real_bool(aa_nat_fun_real_bool(P,I2),aa_nat_real(nth_real(Xs4),I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_4495_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
         => ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I2),X_13)) )
    <=> ? [Xs4: list_bool] :
          ( ( size_size_list_bool(Xs4) = K )
          & ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I2),aa_nat_bool(nth_bool(Xs4),I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_4496_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),X_13)) )
    <=> ? [Xs4: list_nat] :
          ( ( size_size_list_nat(Xs4) = K )
          & ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(nth_nat(Xs4),I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_4497_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
         => ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I2),X_13)) )
    <=> ? [Xs4: list_int] :
          ( ( size_size_list_int(Xs4) = K )
          & ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),K))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I2),aa_nat_int(nth_int(Xs4),I2))) ) ) ) ).

% Skolem_list_nth
tff(fact_4498_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4499_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
           => ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4500_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
           => ( aa_nat_real(nth_real(Xs),I3) = aa_nat_real(nth_real(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4501_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I3))
            <=> pp(aa_nat_bool(nth_bool(Ys),I3)) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4502_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4503_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4504_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE3763956218935105199t_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ? [Li: vEBT_VEBT] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(P,Li),I3)) )
     => ~ ! [L3: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(P,nth_VEBT_VEBT(L3,I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_4505_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE3025410955681380100t_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ? [Li: vEBT_VEBTi] : pp(aa_nat_bool(aa_VEB8502397618584205677t_bool(P,Li),I3)) )
     => ~ ! [L3: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_VEB8502397618584205677t_bool(P,nth_VEBT_VEBTi(L3,I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_4506_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_re9121486324101951821t_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ? [Li: real] : pp(aa_nat_bool(aa_real_fun_nat_bool(P,Li),I3)) )
     => ~ ! [L3: list_real] :
            ( ( size_size_list_real(L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_real_fun_nat_bool(P,aa_nat_real(nth_real(L3),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_4507_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_bo3288154511880838028t_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ? [Li: bool] : pp(aa_nat_bool(aa_bool_fun_nat_bool(P,Li),I3)) )
     => ~ ! [L3: list_bool] :
            ( ( size_size_list_bool(L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_bool_fun_nat_bool(P,aa_nat_bool(nth_bool(L3),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_4508_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ? [Li: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,Li),I3)) )
     => ~ ! [L3: list_nat] :
            ( ( size_size_list_nat(L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(L3),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_4509_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_int_fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => ? [Li: int] : pp(aa_nat_bool(aa_int_fun_nat_bool(P,Li),I3)) )
     => ~ ! [L3: list_int] :
            ( ( size_size_list_int(L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
                 => pp(aa_nat_bool(aa_int_fun_nat_bool(P,aa_nat_int(nth_int(L3),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_4510_less__option__None__Some,axiom,
    ! [X: nat] : pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),some_nat(X))) ).

% less_option_None_Some
tff(fact_4511_less__option__None__Some,axiom,
    ! [X: num] : pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,none_num),some_num(X))) ).

% less_option_None_Some
tff(fact_4512_less__option__None__is__Some,axiom,
    ! [X: option_nat] :
      ( pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(ord_less_option_nat,none_nat),X))
     => ? [Z5: nat] : X = some_nat(Z5) ) ).

% less_option_None_is_Some
tff(fact_4513_less__option__None__is__Some,axiom,
    ! [X: option_num] :
      ( pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(ord_less_option_num,none_num),X))
     => ? [Z5: num] : X = some_num(Z5) ) ).

% less_option_None_is_Some
tff(fact_4514_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( ! [X5: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X5,set_VEBT_VEBTi2(Xs))
         => pp(aa_VEBT_VEBTi_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_4515_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_4516_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_bool] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(Xs)))
         => pp(aa_real_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_4517_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X5: bool] :
          ( member_bool(X5,set_bool2(Xs))
         => pp(aa_bool_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_4518_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_4519_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(Xs)))
         => pp(aa_int_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) ) ) ).

% all_set_conv_all_nth
tff(fact_4520_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool,X: vEBT_VEBTi] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
         => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,I3))) )
     => ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
       => pp(aa_VEBT_VEBTi_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4521_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,X: complex] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3451745648224563538omplex(Xs)))
         => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),I3))) )
     => ( member_complex(X,set_complex2(Xs))
       => pp(aa_complex_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4522_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3))) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4523_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: real] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I3))) )
     => ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4524_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X: bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I3))) )
     => ( member_bool(X,set_bool2(Xs))
       => pp(aa_bool_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4525_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4526_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_4527_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs)))
          & ( nth_VEBT_VEBTi(Xs,I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_4528_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3451745648224563538omplex(Xs)))
          & ( aa_nat_complex(nth_complex(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_4529_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_4530_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_4531_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I2))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_4532_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_4533_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I2) = X ) ) ) ).

% in_set_conv_nth
tff(fact_4534_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( ! [X2: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
           => pp(aa_VEBT_VEBTi_bool(P,X2)) )
       => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_4535_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
           => pp(aa_VEBT_VEBT_bool(P,X2)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_4536_list__ball__nth,axiom,
    ! [N: nat,Xs: list_real,P: fun_real_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
           => pp(aa_real_bool(P,X2)) )
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_4537_list__ball__nth,axiom,
    ! [N: nat,Xs: list_bool,P: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => ( ! [X2: bool] :
            ( member_bool(X2,set_bool2(Xs))
           => pp(aa_bool_bool(P,X2)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_4538_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_4539_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
           => pp(aa_int_bool(P,X2)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_4540_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => member_VEBT_VEBTi(nth_VEBT_VEBTi(Xs,N),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_4541_nth__mem,axiom,
    ! [N: nat,Xs: list_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => member_complex(aa_nat_complex(nth_complex(Xs),N),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_4542_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,N),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_4543_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).

% nth_mem
tff(fact_4544_nth__mem,axiom,
    ! [N: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(aa_nat_bool(nth_bool(Xs),N),set_bool2(Xs)) ) ).

% nth_mem
tff(fact_4545_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_4546_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_4547_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_bool] :
      ( ! [X5: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X5,set_VEBT_VEBTi2(L))
         => pp(aa_VEBT_VEBTi_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s7982070591426661849_VEBTi(L)))
         => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(L,I2))) ) ) ).

% all_set_conv_nth
tff(fact_4548_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(L))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(L)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(L,I2))) ) ) ).

% all_set_conv_nth
tff(fact_4549_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_bool] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(L)))
         => pp(aa_real_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(L)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(L),I2))) ) ) ).

% all_set_conv_nth
tff(fact_4550_all__set__conv__nth,axiom,
    ! [L: list_bool,P: fun_bool_bool] :
      ( ! [X5: bool] :
          ( member_bool(X5,set_bool2(L))
         => pp(aa_bool_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(L)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(L),I2))) ) ) ).

% all_set_conv_nth
tff(fact_4551_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(L)))
         => pp(aa_nat_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(L)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(L),I2))) ) ) ).

% all_set_conv_nth
tff(fact_4552_all__set__conv__nth,axiom,
    ! [L: list_int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(L)))
         => pp(aa_int_bool(P,X5)) )
    <=> ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(L)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(L),I2))) ) ) ).

% all_set_conv_nth
tff(fact_4553_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1571388752190420990T_assn,A5: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                        & ( nth_VEBT_VEBT(Xsi,I3) = nth_VEBT_VEBT(Xsi2,I3) ) ) ) ) )
             => ( vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4554_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE5577048379444288083T_assn,A5: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                        & ( nth_VEBT_VEBT(Xsi,I3) = nth_VEBT_VEBT(Xsi2,I3) ) ) ) ) )
             => ( vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) = vEBT_L2497118539674116125T_VEBT(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4555_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE2561185598974572172i_assn,A5: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
         => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                     => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                        & ( nth_VEBT_VEBTi(Xsi,I3) = nth_VEBT_VEBTi(Xsi2,I3) ) ) ) ) )
             => ( vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) = vEBT_L886525131989349516_VEBTi(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4556_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE6910866152645720856l_assn,A5: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Xsi) )
                     => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                        & ( aa_nat_real(nth_real(Xsi),I3) = aa_nat_real(nth_real(Xsi2),I3) ) ) ) ) )
             => ( vEBT_L4281036506115550016T_real(I5,A2,Xs,Xsi) = vEBT_L4281036506115550016T_real(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4557_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1885522831824361987l_assn,A5: fun_VE1885522831824361987l_assn,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_real(Xsi) )
                     => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                        & ( aa_nat_real(nth_real(Xsi),I3) = aa_nat_real(nth_real(Xsi2),I3) ) ) ) ) )
             => ( vEBT_L7728200936804140803i_real(I5,A2,Xs,Xsi) = vEBT_L7728200936804140803i_real(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4558_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1323945509527881313l_assn,A5: fun_VE1323945509527881313l_assn,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_bool,Xsi2: list_bool] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
         => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Xsi) )
                     => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                        & ( pp(aa_nat_bool(nth_bool(Xsi),I3))
                        <=> pp(aa_nat_bool(nth_bool(Xsi2),I3)) ) ) ) ) )
             => ( vEBT_L6176004506706552451T_bool(I5,A2,Xs,Xsi) = vEBT_L6176004506706552451T_bool(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4559_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE3808733824349961398l_assn,A5: fun_VE3808733824349961398l_assn,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_bool,Xsi2: list_bool] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
         => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_bool(Xsi) )
                     => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                        & ( pp(aa_nat_bool(nth_bool(Xsi),I3))
                        <=> pp(aa_nat_bool(nth_bool(Xsi2),I3)) ) ) ) ) )
             => ( vEBT_L2302322349616425472i_bool(I5,A2,Xs,Xsi) = vEBT_L2302322349616425472i_bool(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4560_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE4931415397359990460t_assn,A5: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Xsi) )
                     => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                        & ( aa_nat_nat(nth_nat(Xsi),I3) = aa_nat_nat(nth_nat(Xsi2),I3) ) ) ) ) )
             => ( vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4561_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE1225457145571040039t_assn,A5: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_nat(Xsi) )
                     => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                        & ( aa_nat_nat(nth_nat(Xsi),I3) = aa_nat_nat(nth_nat(Xsi2),I3) ) ) ) ) )
             => ( vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi) = vEBT_L2809031099982602151Ti_nat(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4562_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: fun_VE123837653852366616t_assn,A5: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A2 = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
         => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
           => ( ! [I3: nat] :
                  ( pp(aa_set_nat_bool(member_nat(I3),I5))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Xsi) )
                     => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                        & ( aa_nat_int(nth_int(Xsi),I3) = aa_nat_int(nth_int(Xsi2),I3) ) ) ) ) )
             => ( vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi) = vEBT_L8648204552663881920BT_int(I6,A5,Xs3,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_4563_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,A5: fun_VE1571388752190420990T_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                      & ( nth_VEBT_VEBT(Xsi,I3) = nth_VEBT_VEBT(Xsi2,I3) )
                      & ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I3)),nth_VEBT_VEBT(Xsi,I3)) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A5,nth_VEBT_VEBT(Xs3,I3)),nth_VEBT_VEBT(Xsi2,I3)) ) ) ) ) )
           => ( vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4564_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_VE5577048379444288083T_assn,A5: fun_VE5577048379444288083T_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                      & ( nth_VEBT_VEBT(Xsi,I3) = nth_VEBT_VEBT(Xsi2,I3) )
                      & ( aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I3)),nth_VEBT_VEBT(Xsi,I3)) = aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A5,nth_VEBT_VEBTi(Xs3,I3)),nth_VEBT_VEBT(Xsi2,I3)) ) ) ) ) )
           => ( vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) = vEBT_L2497118539674116125T_VEBT(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4565_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,A5: fun_VE2561185598974572172i_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
       => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                   => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                      & ( nth_VEBT_VEBTi(Xsi,I3) = nth_VEBT_VEBTi(Xsi2,I3) )
                      & ( aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I3)),nth_VEBT_VEBTi(Xsi,I3)) = aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A5,nth_VEBT_VEBTi(Xs3,I3)),nth_VEBT_VEBTi(Xsi2,I3)) ) ) ) ) )
           => ( vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) = vEBT_L886525131989349516_VEBTi(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4566_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: fun_VE6910866152645720856l_assn,A5: fun_VE6910866152645720856l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Xsi) )
                   => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                      & ( aa_nat_real(nth_real(Xsi),I3) = aa_nat_real(nth_real(Xsi2),I3) )
                      & ( aa_real_assn(aa_VEB8575627266564790985l_assn(A2,nth_VEBT_VEBT(Xs,I3)),aa_nat_real(nth_real(Xsi),I3)) = aa_real_assn(aa_VEB8575627266564790985l_assn(A5,nth_VEBT_VEBT(Xs3,I3)),aa_nat_real(nth_real(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L4281036506115550016T_real(I5,A2,Xs,Xsi) = vEBT_L4281036506115550016T_real(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4567_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real,A2: fun_VE1885522831824361987l_assn,A5: fun_VE1885522831824361987l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_real(Xsi) )
                   => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                      & ( aa_nat_real(nth_real(Xsi),I3) = aa_nat_real(nth_real(Xsi2),I3) )
                      & ( aa_real_assn(aa_VEB9107798209145535940l_assn(A2,nth_VEBT_VEBTi(Xs,I3)),aa_nat_real(nth_real(Xsi),I3)) = aa_real_assn(aa_VEB9107798209145535940l_assn(A5,nth_VEBT_VEBTi(Xs3,I3)),aa_nat_real(nth_real(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L7728200936804140803i_real(I5,A2,Xs,Xsi) = vEBT_L7728200936804140803i_real(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4568_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_bool,Xsi2: list_bool,A2: fun_VE1323945509527881313l_assn,A5: fun_VE1323945509527881313l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
       => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Xsi) )
                   => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                      & ( pp(aa_nat_bool(nth_bool(Xsi),I3))
                      <=> pp(aa_nat_bool(nth_bool(Xsi2),I3)) )
                      & ( aa_bool_assn(aa_VEB2879522196611654106l_assn(A2,nth_VEBT_VEBT(Xs,I3)),aa_nat_bool(nth_bool(Xsi),I3)) = aa_bool_assn(aa_VEB2879522196611654106l_assn(A5,nth_VEBT_VEBT(Xs3,I3)),aa_nat_bool(nth_bool(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L6176004506706552451T_bool(I5,A2,Xs,Xsi) = vEBT_L6176004506706552451T_bool(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4569_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_bool,Xsi2: list_bool,A2: fun_VE3808733824349961398l_assn,A5: fun_VE3808733824349961398l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
       => ( ( size_size_list_bool(Xsi) = size_size_list_bool(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_bool(Xsi) )
                   => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                      & ( pp(aa_nat_bool(nth_bool(Xsi),I3))
                      <=> pp(aa_nat_bool(nth_bool(Xsi2),I3)) )
                      & ( aa_bool_assn(aa_VEB1972364096475303967l_assn(A2,nth_VEBT_VEBTi(Xs,I3)),aa_nat_bool(nth_bool(Xsi),I3)) = aa_bool_assn(aa_VEB1972364096475303967l_assn(A5,nth_VEBT_VEBTi(Xs3,I3)),aa_nat_bool(nth_bool(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L2302322349616425472i_bool(I5,A2,Xs,Xsi) = vEBT_L2302322349616425472i_bool(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4570_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE4931415397359990460t_assn,A5: fun_VE4931415397359990460t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Xsi) )
                   => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                      & ( aa_nat_nat(nth_nat(Xsi),I3) = aa_nat_nat(nth_nat(Xsi2),I3) )
                      & ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I3)),aa_nat_nat(nth_nat(Xsi),I3)) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A5,nth_VEBT_VEBT(Xs3,I3)),aa_nat_nat(nth_nat(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4571_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs3: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE1225457145571040039t_assn,A5: fun_VE1225457145571040039t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs3) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs)))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_nat(Xsi) )
                   => ( ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Xs3,I3) )
                      & ( aa_nat_nat(nth_nat(Xsi),I3) = aa_nat_nat(nth_nat(Xsi2),I3) )
                      & ( aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I3)),aa_nat_nat(nth_nat(Xsi),I3)) = aa_nat_assn(aa_VEB8068455526268438248t_assn(A5,nth_VEBT_VEBTi(Xs3,I3)),aa_nat_nat(nth_nat(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi) = vEBT_L2809031099982602151Ti_nat(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4572_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: fun_VE123837653852366616t_assn,A5: fun_VE123837653852366616t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs3) )
       => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Xsi) )
                   => ( ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Xs3,I3) )
                      & ( aa_nat_int(nth_int(Xsi),I3) = aa_nat_int(nth_int(Xsi2),I3) )
                      & ( aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I3)),aa_nat_int(nth_int(Xsi),I3)) = aa_int_assn(aa_VEB392321844297299145t_assn(A5,nth_VEBT_VEBT(Xs3,I3)),aa_nat_int(nth_int(Xsi2),I3)) ) ) ) ) )
           => ( vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi) = vEBT_L8648204552663881920BT_int(I6,A5,Xs3,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_4573_option_Osize__gen_I1_J,axiom,
    ! [X: fun_nat_nat] : size_option_nat(X,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4574_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4575_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4576_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_4577_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_4578_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_4579_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),X) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_4580_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),X) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_4581_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_4582_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_4583_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_4584_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_4585_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_4586_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_4587_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_le
tff(fact_4588_not__Some__eq2,axiom,
    ! [V: option936205604648967762et_nat] :
      ( ! [X5: heap_e7401611519738050253t_unit,Y6: set_nat] : V != some_P624177172695371229et_nat(produc7507926704131184380et_nat(X5,Y6))
    <=> ( V = none_P533106815845188193et_nat ) ) ).

% not_Some_eq2
tff(fact_4589_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ! [X5: num,Y6: num] : V != some_P6201964756284913402um_num(product_Pair_num_num(X5,Y6))
    <=> ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
tff(fact_4590_not__Some__eq2,axiom,
    ! [V: option642762832853965969at_num] :
      ( ! [X5: nat,Y6: num] : V != some_P8071634352977444016at_num(product_Pair_nat_num(X5,Y6))
    <=> ( V = none_P6264349658649815852at_num ) ) ).

% not_Some_eq2
tff(fact_4591_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X5: int,Y6: int] : V != some_P4184893108420464158nt_int(product_Pair_int_int(X5,Y6))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_4592_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X5: nat,Y6: nat] : V != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y6))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_4593_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_4594_log__eq__one,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_4595_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_4596_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_4597_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_4598_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_4599_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_4600_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_4601_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).

% less_log_of_power
tff(fact_4602_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_4603_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_4604_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).

% le_log_of_power
tff(fact_4605_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_4606_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_4607_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_nat_power
tff(fact_4608_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
     => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_4609_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_less
tff(fact_4610_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% log_of_power_le
tff(fact_4611_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_4612_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_4613_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% log2_of_power_less
tff(fact_4614_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),N))
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% nested_mint
tff(fact_4615_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_4616_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_4617_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_4618_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4619_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_4620_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_4621_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_4622_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_4623_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
             => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) )
          & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_4624_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_4625_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_4626_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4627_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4628_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_4629_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_4630_option_Osel,axiom,
    ! [X22: nat] : the_nat(some_nat(X22)) = X22 ).

% option.sel
tff(fact_4631_option_Osel,axiom,
    ! [X22: num] : the_num(some_num(X22)) = X22 ).

% option.sel
tff(fact_4632_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4633_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4634_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4635_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4636_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4637_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4638_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_4639_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) )
                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_4640_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X5)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_4641_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),X))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_4642_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = N )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I3: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_4643_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I3: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_1)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I3 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I3 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X2))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_4644_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_4645_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_4646_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).

% in_children_def
tff(fact_4647_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg))
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% summaxma
tff(fact_4648_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat)))) ) ).

% inrange
tff(fact_4649_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% delt_out_of_range
tff(fact_4650_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary) ).

% vebt_insert.simps(4)
tff(fact_4651_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_vebt_delete(T,X)),N)) ) ).

% delete_pres_valid
tff(fact_4652_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_4653_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( N = one_one_nat )
       => ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ) ).

% deg_1_Leafy
tff(fact_4654_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
     => ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ).

% deg_1_Leaf
tff(fact_4655_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),one_one_nat))
    <=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).

% deg1Leaf
tff(fact_4656_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(X) )
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X)) ) ).

% maxbmo
tff(fact_4657_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),Y)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_4658_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_4659_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),Maxi)) ) ) ).

% maxt_member
tff(fact_4660_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_4661_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_4662_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).

% maxt_sound
tff(fact_4663_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% maxt_corr
tff(fact_4664_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ) ) ) ).

% del_single_cont
tff(fact_4665_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_4666_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X222: bool] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_4667_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142)
     => ~ ! [X212: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_4668_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,Uv: bool,D3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),D3)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_4669_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D)))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4670_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))
    <=> ( ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
          | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
            & ( pp(aa_set_nat_bool(ord_less_set_nat(C),A))
              | pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4671_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D)))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4672_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4673_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D)))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4674_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C,D)))
    <=> ( ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
          | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
            & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),D))
            & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A))
              | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),D)) ) ) )
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4675_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D)))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4676_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_nat_bool(P,M5)) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less
tff(fact_4677_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
          & pp(aa_nat_bool(P,M5)) )
    <=> ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less
tff(fact_4678_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_4679_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv: bool] : X != vEBT_Leaf(fTrue,Uv)
       => ( ! [Uu: bool] : X != vEBT_Leaf(Uu,fTrue)
         => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)
           => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_4680_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: bool,B: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(fTrue,B) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_4681_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_4682_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B4: bool,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X2)
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw),Ux)
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_4683_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_4684_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A4: bool,B4: bool] : X != vEBT_Leaf(A4,B4)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_4685_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_4686_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Leaf(A,B)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_4687_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf(A,B)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_4688_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: bool,Uw2: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_4689_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Leaf(A,B)),zero_zero_nat) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_4690_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf(A,B)),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_4691_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: bool,Uv2: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu2,Uv2),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_4692_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or4029947393144176647an_rat(C,D)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4693_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or3540276404033026485et_nat(C,D))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
          & pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4694_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or1222409239386451017an_num(C,D)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4695_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or66887138388493659n_real(C,D)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4696_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(C),D)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4697_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or4662586982721622107an_int(C,D)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4698_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A,B)),set_or8404916559141939852nteger(C,D)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),D)) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4699_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B)),set_or633870826150836451st_rat(C,D)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4700_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_or66887138388493659n_real(A,B)),set_or1222579329274155063t_real(C,D)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4701_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A: bool,B: bool] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_4702_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A: bool,B: bool] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_4703_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Leaf(A,B)) = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_4704_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Leaf(A,B)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_4705_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,Uv: bool,Uw: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Uw)
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Uz)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb),X2)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),X2)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_4706_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,Uv: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),zero_zero_nat)
     => ( ! [A4: bool,Uw: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,Uw),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: bool,B4: bool,Va: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,Va)))
         => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3),Vb)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X2) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_4707_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: bool,B4: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu,B4),zero_zero_nat)
     => ( ! [Uv: bool,Uw: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N2))
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va3: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X2) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_4708_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B4: bool,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X2)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),X2)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),X2)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X2) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_4709_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B4: bool,X2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X2)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),X2)
       => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),X2)
         => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X2) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_4710_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A4: bool,B4: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),zero_zero_nat)
     => ( ! [A4: bool,B4: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: bool,B4: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,N2)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Uu)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),X2)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),X2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X2) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_4711_vebt__pred_Osimps_I2_J,axiom,
    ! [A: bool,Uw2: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw2),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw2),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_4712_vebt__succ_Osimps_I1_J,axiom,
    ! [B: bool,Uu2: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu2,B),zero_zero_nat) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu2,B),zero_zero_nat) = none_nat ) ) ) ).

% vebt_succ.simps(1)
tff(fact_4713_vebt__pred_Osimps_I3_J,axiom,
    ! [B: bool,A: bool,Va2: nat] :
      ( ( pp(B)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_4714_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S) ).

% vebt_insert.simps(2)
tff(fact_4715_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S) ).

% vebt_insert.simps(3)
tff(fact_4716_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
     => ( ( ? [A4: bool,B4: bool] : A1 = vEBT_Leaf(A4,B4)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N2)) )
                 => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                         => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N2)) )
                   => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N2) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                           => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N2)) )
                     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                             => ( ! [I4: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N2) = I4 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N2))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X3),N2)) )
                       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary2),M3))
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N2) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                               => ( ! [I4: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M3)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I4 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N2) = I4 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N2))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X3))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_4717_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,A1),A22))
    <=> ( ( ? [A6: bool,B6: bool] : A1 = vEBT_Leaf(A6,B6)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N7))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),N7) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N7)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N7)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),aa_nat_nat(suc,N7)) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X5: vEBT_VEBT] :
                ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),N7))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),N7) )
            & ! [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I2)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N7) = I2 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N7))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N7) = I2 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N7))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N7: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X5),N7)) )
            & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N7)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N7)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N7),aa_nat_nat(suc,N7)) )
            & ! [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N7))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I2)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X5),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,N7))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N7) = I2 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N7))) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N7) = I2 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N7))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X5))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X5),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_4718_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( pp(B4)
                 => ( Y = some_nat(one_one_nat) ) )
                & ( ~ pp(B4)
                 => ( ( pp(A4)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A4)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_4719_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( pp(A4)
                 => ( Y = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A4)
                 => ( ( pp(B4)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B4)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_4720_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(T,X)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one)))))) ) ) ).

% minNull_delete_time_bound
tff(fact_4721_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: bool,A: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_4722_vebt__mint_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( ( pp(B)
           => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
          & ( ~ pp(B)
           => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_4723_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) ).

% vebt_delete.simps(6)
tff(fact_4724_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz2)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_4725_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1)) ) ).

% not_min_Null_member
tff(fact_4726_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T),X)) ) ).

% min_Null_member
tff(fact_4727_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_4728_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_4729_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(T,X)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),X)),one_one_nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_4730_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(plus_plus_int(U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_4731_bset_I1_J,axiom,
    ! [D4: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D4))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),D4))) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X3))
                & pp(aa_int_bool(Q,X3)) )
             => ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X3),D4)))
                & pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X3),D4))) ) ) ) ) ) ).

% bset(1)
tff(fact_4732_bset_I2_J,axiom,
    ! [D4: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D4))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X2),D4))) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X3))
                | pp(aa_int_bool(Q,X3)) )
             => ( pp(aa_int_bool(P,aa_int_int(minus_minus_int(X3),D4)))
                | pp(aa_int_bool(Q,aa_int_int(minus_minus_int(X3),D4))) ) ) ) ) ) ).

% bset(2)
tff(fact_4733_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D4))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X2),D4))) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X3))
                & pp(aa_int_bool(Q,X3)) )
             => ( pp(aa_int_bool(P,aa_int_int(plus_plus_int(X3),D4)))
                & pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X3),D4))) ) ) ) ) ) ).

% aset(1)
tff(fact_4734_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D4))) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X2),D4))) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( pp(aa_int_bool(P,X3))
                | pp(aa_int_bool(Q,X3)) )
             => ( pp(aa_int_bool(P,aa_int_int(plus_plus_int(X3),D4)))
                | pp(aa_int_bool(Q,aa_int_int(plus_plus_int(X3),D4))) ) ) ) ) ) ).

% aset(2)
tff(fact_4735_aset_I10_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X3),D4)),T))) ) ) ) ).

% aset(10)
tff(fact_4736_aset_I9_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X3),D4)),T))) ) ) ) ).

% aset(9)
tff(fact_4737_bset_I10_J,axiom,
    ! [D: int,D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X3),D4)),T))) ) ) ) ).

% bset(10)
tff(fact_4738_bset_I9_J,axiom,
    ! [D: int,D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(X3),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X3),D4)),T))) ) ) ) ).

% bset(9)
tff(fact_4739_bset_I3_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(minus_minus_int(T),one_one_int)),B2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( X3 = T )
             => ( aa_int_int(minus_minus_int(X3),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_4740_bset_I4_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( X3 != T )
             => ( aa_int_int(minus_minus_int(X3),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_4741_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(X3),D4)),T)) ) ) ) ).

% bset(5)
tff(fact_4742_bset_I7_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(minus_minus_int(X3),D4))) ) ) ) ) ).

% bset(7)
tff(fact_4743_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(plus_plus_int(T),one_one_int)),A2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( X3 = T )
             => ( aa_int_int(plus_plus_int(X3),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_4744_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( X3 != T )
             => ( aa_int_int(plus_plus_int(X3),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_4745_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(plus_plus_int(X3),D4)),T)) ) ) ) ) ).

% aset(5)
tff(fact_4746_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(plus_plus_int(X3),D4))) ) ) ) ).

% aset(7)
tff(fact_4747_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X2: int,K2: int] :
            ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
        <=> ? [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),set_or1266510415728281911st_int(one_one_int,D)))
              & pp(aa_int_bool(P,X5)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4748_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(X3),D4)),T)) ) ) ) ).

% bset(6)
tff(fact_4749_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(minus_minus_int(T),one_one_int)),B2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(minus_minus_int(X3),D4))) ) ) ) ) ).

% bset(8)
tff(fact_4750_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(plus_plus_int(T),one_one_int)),A2))
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(X3),D4)),T)) ) ) ) ) ).

% aset(6)
tff(fact_4751_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( pp(aa_set_int_bool(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(plus_plus_int(X3),D4))) ) ) ) ).

% aset(8)
tff(fact_4752_cppi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z4),X2))
           => ( pp(aa_int_bool(P,X2))
            <=> pp(aa_int_bool(P2,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),A2))
                     => ( X2 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X2))
               => pp(aa_int_bool(P,aa_int_int(plus_plus_int(X2),D4))) ) )
         => ( ! [X2: int,K2: int] :
                ( pp(aa_int_bool(P2,X2))
              <=> pp(aa_int_bool(P2,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D4)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4)))
                    & pp(aa_int_bool(P2,X5)) )
                | ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),A2))
                        & pp(aa_int_bool(P,aa_int_int(minus_minus_int(Xa3),X5))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4753_cpmi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z4))
           => ( pp(aa_int_bool(P,X2))
            <=> pp(aa_int_bool(P2,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),B2))
                     => ( X2 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X2))
               => pp(aa_int_bool(P,aa_int_int(minus_minus_int(X2),D4))) ) )
         => ( ! [X2: int,K2: int] :
                ( pp(aa_int_bool(P2,X2))
              <=> pp(aa_int_bool(P2,aa_int_int(minus_minus_int(X2),aa_int_int(times_times_int(K2),D4)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4)))
                    & pp(aa_int_bool(P2,X5)) )
                | ? [X5: int] :
                    ( pp(aa_set_int_bool(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4)))
                    & ? [Xa3: int] :
                        ( pp(aa_set_int_bool(member_int(Xa3),B2))
                        & pp(aa_int_bool(P,aa_int_int(plus_plus_int(Xa3),X5))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4754_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1260151622200656695at_nat] :
      ( ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : X != produc7814092408141220007at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2477239776880934807at_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4755_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc7004362771917097900on_nat] :
      ( ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : X != produc6699827698519632094on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_nat,V2: nat] : X != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_nat,A4: nat,B4: nat] : X != produc6699827698519632094on_nat(F3,produc5098337634421038937on_nat(some_nat(A4),some_nat(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4756_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc2703417854250991842on_num] :
      ( ! [Uu: fun_num_fun_num_num,Uv: option_num] : X != produc6360833703471310868on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_num,V2: num] : X != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F3: fun_num_fun_num_num,A4: num,B4: num] : X != produc6360833703471310868on_num(F3,produc8585076106096196333on_num(some_num(A4),some_num(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4757_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc533602882674032673at_nat] :
      ( ! [Uu: fun_Pr248683899324676269t_bool,Uv: option4927543243414619207at_nat] : X != produc5961576194287291153at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr248683899324676269t_bool,V2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr248683899324676269t_bool,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X2),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4758_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc3700678068921409119on_nat] :
      ( ! [Uu: fun_nat_fun_nat_bool,Uv: option_nat] : X != produc2004789812759622991on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_bool,V2: nat] : X != produc2004789812759622991on_nat(Uw,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_bool,X2: nat,Y3: nat] : X != produc2004789812759622991on_nat(F3,produc5098337634421038937on_nat(some_nat(X2),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4759_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc6167219322466932359on_num] :
      ( ! [Uu: fun_num_fun_num_bool,Uv: option_num] : X != produc8433281988922931063on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_bool,V2: num] : X != produc8433281988922931063on_num(Uw,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F3: fun_num_fun_num_bool,X2: num,Y3: num] : X != produc8433281988922931063on_num(F3,produc8585076106096196333on_num(some_num(X2),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4760_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_4761_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_4762_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_4763_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu2,none_P5556105721700978146at_nat,Uv2) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_4764_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : vEBT_V819420779217536731ft_num(Uu2,none_num,Uv2) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_4765_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu2),none_nat),Uv2) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_4766_vebt__delete_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,N))) = vEBT_Leaf(A,B) ).

% vebt_delete.simps(3)
tff(fact_4767_vebt__delete_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : vEBT_vebt_delete(vEBT_Leaf(A,B),zero_zero_nat) = vEBT_Leaf(fFalse,B) ).

% vebt_delete.simps(1)
tff(fact_4768_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw2,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_4769_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw2,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_4770_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw2),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_4771_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_Pr2477239776880934807at_nat,Xa4: option4927543243414619207at_nat,Xb3: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(X,Xa4,Xb3) = Y )
     => ( ( ( Xa4 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xa4 = some_P7363390416028606310at_nat(V2)
           => ( ( Xb3 = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xa4 = some_P7363390416028606310at_nat(A4) )
               => ! [B4: product_prod_nat_nat] :
                    ( ( Xb3 = some_P7363390416028606310at_nat(B4) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_4772_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_num_fun_num_num,Xa4: option_num,Xb3: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(X,Xa4,Xb3) = Y )
     => ( ( ( Xa4 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xa4 = some_num(V2)
           => ( ( Xb3 = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A4: num] :
                ( ( Xa4 = some_num(A4) )
               => ! [B4: num] :
                    ( ( Xb3 = some_num(B4) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_4773_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa4: option_nat,Xb3: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa4),Xb3) = Y )
     => ( ( ( Xa4 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xa4 = some_nat(V2)
           => ( ( Xb3 = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A4: nat] :
                ( ( Xa4 = some_nat(A4) )
               => ! [B4: nat] :
                    ( ( Xb3 = some_nat(B4) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_4774_vebt__delete_Osimps_I2_J,axiom,
    ! [A: bool,B: bool] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(A,fFalse) ).

% vebt_delete.simps(2)
tff(fact_4775_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry) ).

% vebt_delete.simps(5)
tff(fact_4776_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz2)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_4777_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_4778_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_4779_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv: bool] : X = vEBT_Leaf(fTrue,Uv)
           => pp(Y) )
         => ( ( ? [Uu: bool] : X = vEBT_Leaf(Uu,fTrue)
             => pp(Y) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)
               => ~ pp(Y) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_4780_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))))) ) ) ).

% member_bound_size_univ
tff(fact_4781_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_4782_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% heigt_uplog_rel
tff(fact_4783_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).

% list_update_overwrite
tff(fact_4784_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi,Y: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X),I,Y) = list_u6098035379799741383_VEBTi(Xs,I,Y) ).

% list_update_overwrite
tff(fact_4785_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_4786_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_4787_length__list__update,axiom,
    ! [Xs: list_real,I: nat,X: real] : size_size_list_real(list_update_real(Xs,I,X)) = size_size_list_real(Xs) ).

% length_list_update
tff(fact_4788_length__list__update,axiom,
    ! [Xs: list_bool,I: nat,X: bool] : size_size_list_bool(list_update_bool(Xs,I,X)) = size_size_list_bool(Xs) ).

% length_list_update
tff(fact_4789_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : size_size_list_nat(list_update_nat(Xs,I,X)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_4790_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] : size_size_list_int(list_update_int(Xs,I,X)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_4791_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_4792_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_4793_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_4794_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] : list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,I)) = Xs ).

% list_update_id
tff(fact_4795_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_nat,X: nat] :
      ( ( I != J2 )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J2) = aa_nat_nat(nth_nat(Xs),J2) ) ) ).

% nth_list_update_neq
tff(fact_4796_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_int,X: int] :
      ( ( I != J2 )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J2) = aa_nat_int(nth_int(Xs),J2) ) ) ).

% nth_list_update_neq
tff(fact_4797_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J2 )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J2) = nth_VEBT_VEBT(Xs,J2) ) ) ).

% nth_list_update_neq
tff(fact_4798_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( I != J2 )
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X),J2) = nth_VEBT_VEBTi(Xs,J2) ) ) ).

% nth_list_update_neq
tff(fact_4799_ceiling__zero,axiom,
    archim2889992004027027881ng_rat(zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_4800_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_4801_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4802_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4803_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
     => ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4804_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I))
     => ( list_u6098035379799741383_VEBTi(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4805_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_real(Xs)),I))
     => ( list_update_real(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4806_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I))
     => ( list_update_bool(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4807_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
     => ( list_update_nat(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4808_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I))
     => ( list_update_int(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4809_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J2,X),I) = nth_VEBT_VEBT(L,I) ) ) ).

% nth_update_invalid
tff(fact_4810_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J2: nat,X: vEBT_VEBTi] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J2,X),I) = nth_VEBT_VEBTi(L,I) ) ) ).

% nth_update_invalid
tff(fact_4811_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J2: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( aa_nat_real(nth_real(list_update_real(L,J2,X)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_4812_nth__update__invalid,axiom,
    ! [I: nat,L: list_bool,J2: nat,X: bool] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(L,J2,X)),I))
      <=> pp(aa_nat_bool(nth_bool(L),I)) ) ) ).

% nth_update_invalid
tff(fact_4813_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J2: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J2,X)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_4814_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J2: nat,X: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( aa_nat_int(nth_int(list_update_int(L,J2,X)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_4815_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4816_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4817_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4818_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),I))
      <=> pp(X) ) ) ).

% nth_list_update_eq
tff(fact_4819_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4820_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4821_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_le_zero
tff(fact_4822_ceiling__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_le_zero
tff(fact_4823_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),numeral_numeral_rat(V))) ) ).

% ceiling_le_numeral
tff(fact_4824_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),numeral_numeral_real(V))) ) ).

% ceiling_le_numeral
tff(fact_4825_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% zero_less_ceiling
tff(fact_4826_zero__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_ceiling
tff(fact_4827_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(V)),X)) ) ).

% numeral_less_ceiling
tff(fact_4828_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(V)),X)) ) ).

% numeral_less_ceiling
tff(fact_4829_ceiling__less__one,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),one_one_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).

% ceiling_less_one
tff(fact_4830_ceiling__less__one,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% ceiling_less_one
tff(fact_4831_one__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).

% one_le_ceiling
tff(fact_4832_one__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_le_ceiling
tff(fact_4833_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(V))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4834_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : archim2889992004027027881ng_rat(plus_plus_rat(X,numeral_numeral_rat(V))) = aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4835_one__less__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).

% one_less_ceiling
tff(fact_4836_one__less__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).

% one_less_ceiling
tff(fact_4837_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J2)),J2,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_4838_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_s7982070591426661849_VEBTi(Xs)))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,J2)),J2,nth_VEBT_VEBTi(Xs,I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_4839_set__swap,axiom,
    ! [I: nat,Xs: list_real,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_real(Xs)))
       => ( set_real2(list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J2)),J2,aa_nat_real(nth_real(Xs),I))) = set_real2(Xs) ) ) ) ).

% set_swap
tff(fact_4840_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J2)),J2,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_4841_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J2)),J2,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_4842_set__swap,axiom,
    ! [I: nat,Xs: list_int,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J2)),J2,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_4843_ceiling__add__one,axiom,
    ! [X: rat] : archim2889992004027027881ng_rat(plus_plus_rat(X,one_one_rat)) = aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(X)),one_one_int) ).

% ceiling_add_one
tff(fact_4844_ceiling__add__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_4845_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),numeral_numeral_real(V))) = aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4846_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(X,numeral_numeral_rat(V))) = aa_int_int(minus_minus_int(archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4847_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% ceiling_numeral_power
tff(fact_4848_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).

% ceiling_numeral_power
tff(fact_4849_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),minus_minus_rat(numeral_numeral_rat(V),one_one_rat))) ) ).

% ceiling_less_numeral
tff(fact_4850_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real))) ) ).

% ceiling_less_numeral
tff(fact_4851_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)),X)) ) ).

% numeral_le_ceiling
tff(fact_4852_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),X)) ) ).

% numeral_le_ceiling
tff(fact_4853_list__update__swap,axiom,
    ! [I: nat,I7: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I7 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I7,X6) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I7,X6),I,X) ) ) ).

% list_update_swap
tff(fact_4854_list__update__swap,axiom,
    ! [I: nat,I7: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi,X6: vEBT_VEBTi] :
      ( ( I != I7 )
     => ( list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X),I7,X6) = list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I7,X6),I,X) ) ) ).

% list_update_swap
tff(fact_4855_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,X: complex,I: nat] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),A2)
     => ( member_complex(X,A2)
       => ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_4856_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A2))
     => ( pp(aa_set_real_bool(member_real(X),A2))
       => pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4857_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4858_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( member_VEBT_VEBT(X,A2)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_4859_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,X: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A2)
     => ( member_VEBT_VEBTi(X,A2)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_4860_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4861_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),archim2889992004027027881ng_rat(Y)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_4862_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% ceiling_less_cancel
tff(fact_4863_in__set__upd__eq,axiom,
    ! [I: nat,L: list_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3451745648224563538omplex(L)))
     => ( member_complex(X,set_complex2(list_update_complex(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_complex(X,set_complex2(L))
            & ! [Y6: complex] : member_complex(X,set_complex2(list_update_complex(L,I,Y6))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4864_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_VEBT_VEBT(X,set_VEBT_VEBT2(L))
            & ! [Y6: vEBT_VEBT] : member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y6))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4865_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(L))
            & ! [Y6: vEBT_VEBTi] : member_VEBT_VEBTi(X,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y6))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4866_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(L,I,Y))))
      <=> ( ( X = Y )
          | ( pp(aa_set_real_bool(member_real(X),set_real2(L)))
            & ! [Y6: real] : pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(L,I,Y6)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4867_in__set__upd__eq,axiom,
    ! [I: nat,L: list_bool,X: bool,Y: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( member_bool(X,set_bool2(list_update_bool(L,I,Y)))
      <=> ( ( pp(X)
          <=> pp(Y) )
          | ( member_bool(X,set_bool2(L))
            & ! [Y6: bool] : member_bool(X,set_bool2(list_update_bool(L,I,Y6))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4868_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(L,I,Y))))
      <=> ( ( X = Y )
          | ( pp(aa_set_nat_bool(member_nat(X),set_nat2(L)))
            & ! [Y6: nat] : pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(L,I,Y6)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4869_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(L,I,Y))))
      <=> ( ( X = Y )
          | ( pp(aa_set_int_bool(member_int(X),set_int2(L)))
            & ! [Y6: int] : pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(L,I,Y6)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4870_in__set__upd__cases,axiom,
    ! [X: complex,L: list_complex,I: nat,Y: complex] :
      ( member_complex(X,set_complex2(list_update_complex(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3451745648224563538omplex(L)))
         => ( X != Y ) )
       => member_complex(X,set_complex2(L)) ) ) ).

% in_set_upd_cases
tff(fact_4871_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
         => ( X != Y ) )
       => member_VEBT_VEBT(X,set_VEBT_VEBT2(L)) ) ) ).

% in_set_upd_cases
tff(fact_4872_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
         => ( X != Y ) )
       => member_VEBT_VEBTi(X,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_4873_in__set__upd__cases,axiom,
    ! [X: real,L: list_real,I: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
         => ( X != Y ) )
       => pp(aa_set_real_bool(member_real(X),set_real2(L))) ) ) ).

% in_set_upd_cases
tff(fact_4874_in__set__upd__cases,axiom,
    ! [X: bool,L: list_bool,I: nat,Y: bool] :
      ( member_bool(X,set_bool2(list_update_bool(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
         => ( pp(X)
          <=> ~ pp(Y) ) )
       => member_bool(X,set_bool2(L)) ) ) ).

% in_set_upd_cases
tff(fact_4875_in__set__upd__cases,axiom,
    ! [X: nat,L: list_nat,I: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
         => ( X != Y ) )
       => pp(aa_set_nat_bool(member_nat(X),set_nat2(L))) ) ) ).

% in_set_upd_cases
tff(fact_4876_in__set__upd__cases,axiom,
    ! [X: int,L: list_int,I: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
         => ( X != Y ) )
       => pp(aa_set_int_bool(member_int(X),set_int2(L))) ) ) ).

% in_set_upd_cases
tff(fact_4877_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3451745648224563538omplex(L)))
     => ( member_complex(X,set_complex2(list_update_complex(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y6: complex] : member_complex(X,set_complex2(list_update_complex(L,I,Y6))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4878_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y6: vEBT_VEBT] : member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y6))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4879_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y6: vEBT_VEBTi] : member_VEBT_VEBTi(X,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y6))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4880_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(L,I,Y))))
      <=> ( ( X = Y )
          | ! [Y6: real] : pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(L,I,Y6)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4881_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_bool,X: bool,Y: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( member_bool(X,set_bool2(list_update_bool(L,I,Y)))
      <=> ( ( pp(X)
          <=> pp(Y) )
          | ! [Y6: bool] : member_bool(X,set_bool2(list_update_bool(L,I,Y6))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4882_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(L,I,Y))))
      <=> ( ( X = Y )
          | ! [Y6: nat] : pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(L,I,Y6)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4883_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(L,I,Y))))
      <=> ( ( X = Y )
          | ! [Y6: int] : pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(L,I,Y6)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4884_set__update__memI,axiom,
    ! [N: nat,Xs: list_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
     => member_complex(X,set_complex2(list_update_complex(Xs,N,X))) ) ).

% set_update_memI
tff(fact_4885_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X))) ) ).

% set_update_memI
tff(fact_4886_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => member_VEBT_VEBTi(X,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,N,X))) ) ).

% set_update_memI
tff(fact_4887_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4888_set__update__memI,axiom,
    ! [N: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
     => member_bool(X,set_bool2(list_update_bool(Xs,N,X))) ) ).

% set_update_memI
tff(fact_4889_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4890_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4891_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( ( I = J2 )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))) )
       => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,X),J2) = X ) )
      & ( ~ ( ( I = J2 )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))) )
       => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,X),J2) = nth_VEBT_VEBT(L,J2) ) ) ) ).

% nth_list_update'
tff(fact_4892_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ( ( I = J2 )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))) )
       => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,X),J2) = X ) )
      & ( ~ ( ( I = J2 )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))) )
       => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,X),J2) = nth_VEBT_VEBTi(L,J2) ) ) ) ).

% nth_list_update'
tff(fact_4893_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_real,X: real] :
      ( ( ( ( I = J2 )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L))) )
       => ( aa_nat_real(nth_real(list_update_real(L,I,X)),J2) = X ) )
      & ( ~ ( ( I = J2 )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L))) )
       => ( aa_nat_real(nth_real(list_update_real(L,I,X)),J2) = aa_nat_real(nth_real(L),J2) ) ) ) ).

% nth_list_update'
tff(fact_4894_nth__list__update_H,axiom,
    ! [L: list_bool,I: nat,X: bool,J2: nat] :
      ( pp(aa_nat_bool(nth_bool(list_update_bool(L,I,X)),J2))
    <=> ( ( ( ( I = J2 )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L))) )
         => pp(X) )
        & ( ~ ( ( I = J2 )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L))) )
         => pp(aa_nat_bool(nth_bool(L),J2)) ) ) ) ).

% nth_list_update'
tff(fact_4895_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_nat,X: nat] :
      ( ( ( ( I = J2 )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X)),J2) = X ) )
      & ( ~ ( ( I = J2 )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X)),J2) = aa_nat_nat(nth_nat(L),J2) ) ) ) ).

% nth_list_update'
tff(fact_4896_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_int,X: int] :
      ( ( ( ( I = J2 )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L))) )
       => ( aa_nat_int(nth_int(list_update_int(L,I,X)),J2) = X ) )
      & ( ~ ( ( I = J2 )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L))) )
       => ( aa_nat_int(nth_int(list_update_int(L,I,X)),J2) = aa_nat_int(nth_int(L),J2) ) ) ) ).

% nth_list_update'
tff(fact_4897_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J2 )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J2) = X ) )
        & ( ( I != J2 )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J2) = nth_VEBT_VEBT(Xs,J2) ) ) ) ) ).

% nth_list_update
tff(fact_4898_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J2: nat,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( ( ( I = J2 )
         => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X),J2) = X ) )
        & ( ( I != J2 )
         => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,X),J2) = nth_VEBT_VEBTi(Xs,J2) ) ) ) ) ).

% nth_list_update
tff(fact_4899_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,J2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( ( ( I = J2 )
         => ( aa_nat_real(nth_real(list_update_real(Xs,I,X)),J2) = X ) )
        & ( ( I != J2 )
         => ( aa_nat_real(nth_real(list_update_real(Xs,I,X)),J2) = aa_nat_real(nth_real(Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_4900_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X: bool,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),J2))
      <=> ( ( ( I = J2 )
           => pp(X) )
          & ( ( I != J2 )
           => pp(aa_nat_bool(nth_bool(Xs),J2)) ) ) ) ) ).

% nth_list_update
tff(fact_4901_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J2 )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J2) = X ) )
        & ( ( I != J2 )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J2) = aa_nat_nat(nth_nat(Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_4902_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J2: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( ( I = J2 )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J2) = X ) )
        & ( ( I != J2 )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J2) = aa_nat_int(nth_int(Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_4903_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4904_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBTi(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4905_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
     => ( ( list_update_real(Xs,I,X) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4906_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I,X) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X) ) ) ) ).

% list_update_same_conv
tff(fact_4907_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4908_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4909_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(plus_plus_rat(X,Y))),aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(X)),archim2889992004027027881ng_rat(Y)))) ).

% ceiling_add_le
tff(fact_4910_ceiling__add__le,axiom,
    ! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).

% ceiling_add_le
tff(fact_4911_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(I5,A2,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4912_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4913_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4914_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(I5,A2,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4915_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(I5,A2,list_update_real(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4916_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo5394993412183903963T_assn,X1: bool,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( vEBT_L2992182275864018813T_VEBT(I5,A2,list_update_bool(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L2992182275864018813T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4917_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo7535889243153246212i_assn,X1: bool,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( vEBT_L6372884366951783724_VEBTi(I5,A2,list_update_bool(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = vEBT_L6372884366951783724_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4918_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L8511957252848910786T_VEBT(I5,A2,list_update_nat(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L8511957252848910786T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4919_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L7489483478785760935_VEBTi(I5,A2,list_update_nat(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = vEBT_L7489483478785760935_VEBTi(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4920_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A2: fun_in7755481285845205148T_assn,X1: int,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( vEBT_L2018189785592951398T_VEBT(I5,A2,list_update_int(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L2018189785592951398T_VEBT(I5,A2,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_4921_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B))),aa_int_int(times_times_int(archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).

% mult_ceiling_le
tff(fact_4922_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(times_times_int(aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le
tff(fact_4923_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),X) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_4924_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2),Uz2)),X) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_4925_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_4926_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),X) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_4927_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Leaf(A,B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_4928_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_4929_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_4930_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_4931_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_4932_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2),Uz2)),X)) ).

% vebt_member.simps(3)
tff(fact_4933_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv: bool] : X != vEBT_Leaf(fTrue,Uv)
       => ( ! [Uu: bool] : X != vEBT_Leaf(Uu,fTrue)
         => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_4934_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% member_bound_height
tff(fact_4935_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_4936_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_4937_log__ceil__idem,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),X)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ) ).

% log_ceil_idem
tff(fact_4938_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))))) ) ) ).

% succ_bound_size_univ
tff(fact_4939_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))))) ) ) ).

% pred_bound_size_univ
tff(fact_4940_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))))) ) ) ).

% insert_bound_size_univ
tff(fact_4941_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K6: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4942_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K6: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K6))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K6))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4943_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,W) = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_4944_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W) = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_4945_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,W) = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( W = Z ) ) ).

% of_int_eq_iff
tff(fact_4946_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_real(ring_1_of_int_real,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_real(zero_n3304061248610475627l_real,P) ).

% of_int_of_bool
tff(fact_4947_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_rat(ring_1_of_int_rat,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n2052037380579107095ol_rat(P) ).

% of_int_of_bool
tff(fact_4948_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_complex(ring_17405671764205052669omplex,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n1201886186963655149omplex(P) ).

% of_int_of_bool
tff(fact_4949_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_int(ring_1_of_int_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% of_int_of_bool
tff(fact_4950_of__int__of__bool,axiom,
    ! [P: bool] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n356916108424825756nteger(P) ).

% of_int_of_bool
tff(fact_4951_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)) = X )
    <=> ? [N7: int] : X = aa_int_rat(ring_1_of_int_rat,N7) ) ).

% of_int_ceiling_cancel
tff(fact_4952_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)) = X )
    <=> ? [N7: int] : X = aa_int_real(ring_1_of_int_real,N7) ) ).

% of_int_ceiling_cancel
tff(fact_4953_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_4954_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_4955_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_4956_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_4957_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_4958_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_4959_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_4960_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fdisj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,P)),aa_bool_nat(zero_n2687167440665602831ol_nat,Q)) ).

% of_bool_or_iff
tff(fact_4961_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(P),Q)) = aa_int_int(ord_max_int(aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).

% of_bool_or_iff
tff(fact_4962_of__bool__or__iff,axiom,
    ! [P: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fdisj(P),Q)) = ord_max_Code_integer(zero_n356916108424825756nteger(P),zero_n356916108424825756nteger(Q)) ).

% of_bool_or_iff
tff(fact_4963_max__0__1_I3_J,axiom,
    ! [X: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(3)
tff(fact_4964_max__0__1_I3_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_4965_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).

% max_0_1(3)
tff(fact_4966_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)
tff(fact_4967_max__0__1_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(3)
tff(fact_4968_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(zero_zero_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(3)
tff(fact_4969_max__0__1_I4_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(X) ).

% max_0_1(4)
tff(fact_4970_max__0__1_I4_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_4971_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(X)),zero_zero_real) = numeral_numeral_real(X) ).

% max_0_1(4)
tff(fact_4972_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)
tff(fact_4973_max__0__1_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(4)
tff(fact_4974_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,X)),zero_zero_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(4)
tff(fact_4975_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(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)
tff(fact_4976_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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)
tff(fact_4977_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(1)
tff(fact_4978_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_4979_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_4980_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4981_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = zero_z3403309356797280102nteger )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4982_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4983_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4984_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4985_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4986_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_z3403309356797280102nteger = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4987_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4988_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4989_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4990_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_4991_of__int__0,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,zero_zero_int) = zero_z3403309356797280102nteger ).

% of_int_0
tff(fact_4992_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_4993_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_4994_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_4995_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4996_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4997_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4998_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4999_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_5000_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_5001_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_5002_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_5003_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_5004_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_5005_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_5006_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_5007_max__0__1_I2_J,axiom,
    ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_5008_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_5009_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_5010_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_5011_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_5012_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_5013_max__0__1_I1_J,axiom,
    ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_5014_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_5015_max__0__1_I6_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_5016_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(X)),one_one_real) = numeral_numeral_real(X) ).

% max_0_1(6)
tff(fact_5017_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)
tff(fact_5018_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(6)
tff(fact_5019_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,X)),one_one_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(6)
tff(fact_5020_max__0__1_I5_J,axiom,
    ! [X: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_5021_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).

% max_0_1(5)
tff(fact_5022_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)
tff(fact_5023_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(5)
tff(fact_5024_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(ord_max_int(one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(5)
tff(fact_5025_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_5026_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_5027_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).

% of_int_le_iff
tff(fact_5028_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5029_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5030_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5031_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).

% of_int_less_iff
tff(fact_5032_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5033_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5034_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5035_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5036_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_5037_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_5038_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_5039_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5040_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(plus_plus_int(W),Z)) = aa_int_int(plus_plus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_5041_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_5042_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(W),Z)) = plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_5043_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(plus_plus_int(W),Z)) = aa_complex_complex(plus_plus_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_add
tff(fact_5044_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(times_times_int(W),Z)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_mult
tff(fact_5045_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(times_times_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_5046_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(times_times_int(W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_5047_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(times_times_int(W),Z)) = aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_5048_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(minus_minus_int(W),Z)) = aa_complex_complex(minus_minus_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_diff
tff(fact_5049_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(minus_minus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_5050_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(minus_minus_int(W),Z)) = minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_5051_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(minus_minus_int(W),Z)) = aa_int_int(minus_minus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_5052_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% of_int_of_nat_eq
tff(fact_5053_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% of_int_of_nat_eq
tff(fact_5054_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% of_int_of_nat_eq
tff(fact_5055_of__int__of__nat__eq,axiom,
    ! [N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ).

% of_int_of_nat_eq
tff(fact_5056_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5057_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5058_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5059_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,X) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5060_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5061_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5062_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5063_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5064_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W) = aa_int_Code_integer(ring_18347121197199848620nteger,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5065_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,X) )
    <=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5066_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_5067_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_5068_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),N)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).

% of_int_power
tff(fact_5069_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(power_power_int(Z),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)),N) ).

% of_int_power
tff(fact_5070_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_5071_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] : archim2889992004027027881ng_rat(plus_plus_rat(X,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(X)),Z) ).

% ceiling_add_of_int
tff(fact_5072_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,X)),Z) ).

% ceiling_add_of_int
tff(fact_5073_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5074_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5075_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5076_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).

% of_int_0_le_iff
tff(fact_5077_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5078_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5079_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5080_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_5081_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5082_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5083_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_le_iff
tff(fact_5084_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5085_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5086_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_le_numeral_iff
tff(fact_5087_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5088_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5089_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5090_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_5091_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5092_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5093_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5094_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% of_int_0_less_iff
tff(fact_5095_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(N)),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5096_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5097_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5098_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).

% of_int_numeral_less_iff
tff(fact_5099_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),numera6620942414471956472nteger(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5100_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5101_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5102_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).

% of_int_less_numeral_iff
tff(fact_5103_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5104_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5105_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_5106_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5107_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5108_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).

% of_int_1_le_iff
tff(fact_5109_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5110_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5111_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5112_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_5113_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5114_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5115_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5116_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% of_int_1_less_iff
tff(fact_5117_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5118_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5119_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5120_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5121_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5122_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5123_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5124_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5125_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5126_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5127_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5128_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5129_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5130_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5131_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5132_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5133_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5134_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5135_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5136_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5137_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5138_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5139_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5140_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5141_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5142_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5143_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5144_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5145_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5146_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5147_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5148_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5149_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5150_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5151_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5152_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5153_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5154_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5155_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(X)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5156_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5157_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5158_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5159_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(ord_max_int(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,Y)) ).

% of_int_max
tff(fact_5160_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(ord_max_int(X),Y)) = ord_max_rat(aa_int_rat(ring_1_of_int_rat,X),aa_int_rat(ring_1_of_int_rat,Y)) ).

% of_int_max
tff(fact_5161_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(ord_max_int(X),Y)) = aa_int_int(ord_max_int(aa_int_int(ring_1_of_int_int,X)),aa_int_int(ring_1_of_int_int,Y)) ).

% of_int_max
tff(fact_5162_of__nat__max,axiom,
    ! [X: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = ord_ma741700101516333627d_enat(semiri4216267220026989637d_enat(X),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_5163_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_5164_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_5165_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_5166_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_5167_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : plus_plus_rat(ord_max_rat(X,Y),Z) = ord_max_rat(plus_plus_rat(X,Z),plus_plus_rat(Y,Z)) ).

% max_add_distrib_left
tff(fact_5168_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_5169_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(plus_plus_int(aa_int_int(ord_max_int(X),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(X),Z)),aa_int_int(plus_plus_int(Y),Z)) ).

% max_add_distrib_left
tff(fact_5170_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)) ).

% max_add_distrib_right
tff(fact_5171_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : plus_plus_rat(X,ord_max_rat(Y,Z)) = ord_max_rat(plus_plus_rat(X,Y),plus_plus_rat(X,Z)) ).

% max_add_distrib_right
tff(fact_5172_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% max_add_distrib_right
tff(fact_5173_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(plus_plus_int(X),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(X),Y)),aa_int_int(plus_plus_int(X),Z)) ).

% max_add_distrib_right
tff(fact_5174_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_5175_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : minus_minus_rat(ord_max_rat(X,Y),Z) = ord_max_rat(minus_minus_rat(X,Z),minus_minus_rat(Y,Z)) ).

% max_diff_distrib_left
tff(fact_5176_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(minus_minus_int(aa_int_int(ord_max_int(X),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(minus_minus_int(X),Z)),aa_int_int(minus_minus_int(Y),Z)) ).

% max_diff_distrib_left
tff(fact_5177_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z5: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z5)),X)) ).

% ex_of_int_less
tff(fact_5178_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z5: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z5)),X)) ).

% ex_of_int_less
tff(fact_5179_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z5: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z5))) ).

% ex_less_of_int
tff(fact_5180_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z5: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z5))) ).

% ex_less_of_int
tff(fact_5181_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,X)) ).

% mult_of_int_commute
tff(fact_5182_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_5183_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_5184_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(times_times_int(Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_5185_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_5186_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_5187_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_5188_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_5189_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_5190_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),A)) ) ).

% ceiling_le
tff(fact_5191_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A)) ) ).

% ceiling_le
tff(fact_5192_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_5193_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).

% less_ceiling_iff
tff(fact_5194_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X)))) ).

% real_of_int_div4
tff(fact_5195_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),N))
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_5196_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))) ) ).

% of_int_nonneg
tff(fact_5197_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_nonneg
tff(fact_5198_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_nonneg
tff(fact_5199_of__int__nonneg,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_nonneg
tff(fact_5200_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).

% of_int_pos
tff(fact_5201_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).

% of_int_pos
tff(fact_5202_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).

% of_int_pos
tff(fact_5203_of__int__pos,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))) ) ).

% of_int_pos
tff(fact_5204_floor__exists,axiom,
    ! [X: rat] :
    ? [Z5: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z5)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Z5),one_one_int)))) ) ).

% floor_exists
tff(fact_5205_floor__exists,axiom,
    ! [X: real] :
    ? [Z5: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z5)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Z5),one_one_int)))) ) ).

% floor_exists
tff(fact_5206_floor__exists1,axiom,
    ! [X: rat] :
    ? [X2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(X2),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Y4),one_one_int)))) )
         => ( Y4 = X2 ) ) ) ).

% floor_exists1
tff(fact_5207_floor__exists1,axiom,
    ! [X: real] :
    ? [X2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(X2),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Y4),one_one_int)))) )
         => ( Y4 = X2 ) ) ) ).

% floor_exists1
tff(fact_5208_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2))),plus_plus_rat(R2,one_one_rat))) ).

% of_int_ceiling_le_add_one
tff(fact_5209_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real))) ).

% of_int_ceiling_le_add_one
tff(fact_5210_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2)),one_one_rat)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_5211_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2)) ).

% of_int_ceiling_diff_one_le
tff(fact_5212_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5213_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5214_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5215_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_5216_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real))) ) ).

% int_le_real_less
tff(fact_5217_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).

% int_less_real_le
tff(fact_5218_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(X),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_5219_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S)),X) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_5220_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: bool,Uv2: bool] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf(Uu2,Uv2)),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_5221_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: bool,Uw2: bool,N: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Leaf(Uv2,Uw2)),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_5222_ceiling__split,axiom,
    ! [P: fun_int_bool,T: rat] :
      ( pp(aa_int_bool(P,archim2889992004027027881ng_rat(T)))
    <=> ! [I2: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2),one_one_rat)),T))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I2))) )
         => pp(aa_int_bool(P,I2)) ) ) ).

% ceiling_split
tff(fact_5223_ceiling__split,axiom,
    ! [P: fun_int_bool,T: real] :
      ( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
    <=> ! [I2: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)),T))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I2))) )
         => pp(aa_int_bool(P,I2)) ) ) ).

% ceiling_split
tff(fact_5224_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( archim2889992004027027881ng_rat(X) = A )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))) ) ) ).

% ceiling_eq_iff
tff(fact_5225_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))) ) ) ).

% ceiling_eq_iff
tff(fact_5226_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
       => ( archim2889992004027027881ng_rat(X) = Z ) ) ) ).

% ceiling_unique
tff(fact_5227_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).

% ceiling_unique
tff(fact_5228_ceiling__correct,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)),one_one_rat)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)))) ) ).

% ceiling_correct
tff(fact_5229_ceiling__correct,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).

% ceiling_correct
tff(fact_5230_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),Z))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat))) ) ).

% ceiling_less_iff
tff(fact_5231_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).

% ceiling_less_iff
tff(fact_5232_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X)) ) ).

% le_ceiling_iff
tff(fact_5233_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).

% le_ceiling_iff
tff(fact_5234_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))))) ).

% real_of_int_div2
tff(fact_5235_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_5236_ceiling__divide__upper,axiom,
    ! [Q2: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P3),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P3,Q2)))),Q2))) ) ).

% ceiling_divide_upper
tff(fact_5237_ceiling__divide__upper,axiom,
    ! [Q2: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q2)))),Q2))) ) ).

% ceiling_divide_upper
tff(fact_5238_even__of__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(ring_1_of_int_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)) ) ).

% even_of_int_iff
tff(fact_5239_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S)),X) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_5240_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,Va2: nat] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf(A,B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_5241_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Leaf(A,B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_5242_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_5243_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: bool,B: bool] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Leaf(Uu2,B)),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_5244_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_5245_ceiling__divide__lower,axiom,
    ! [Q2: real,P3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q2)))),one_one_real)),Q2)),P3)) ) ).

% ceiling_divide_lower
tff(fact_5246_ceiling__divide__lower,axiom,
    ! [Q2: rat,P3: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P3,Q2))),one_one_rat)),Q2)),P3)) ) ).

% ceiling_divide_lower
tff(fact_5247_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,N),one_one_rat)))
       => ( archim2889992004027027881ng_rat(X) = aa_int_int(plus_plus_int(N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5248_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(plus_plus_int(N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5249_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: bool,Uw2: bool] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf(A,Uw2)),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_5250_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_5251_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_5252_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),X) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_5253_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ) ) ).

% insersimp
tff(fact_5254_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ) ) ).

% insertsimp
tff(fact_5255_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(one)))))))) ) ).

% insert_bound_height
tff(fact_5256_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one)))))))) ) ).

% pred_bound_height
tff(fact_5257_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))))) ) ).

% succ_bound_height
tff(fact_5258_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5259_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5260_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5261_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5262_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_5263_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_5264_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5265_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5266_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_5267_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_5268_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5269_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5270_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_ma741700101516333627d_enat(Q2,zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_5271_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_5272_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_5273_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_5274_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_5275_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_5276_norm__mult,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_5277_norm__mult,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_5278_norm__not__less__zero,axiom,
    ! [X: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)) ).

% norm_not_less_zero
tff(fact_5279_norm__ge__zero,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% norm_ge_zero
tff(fact_5280_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_5281_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_5282_norm__power,axiom,
    ! [X: real,N: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),N) ).

% norm_power
tff(fact_5283_norm__power,axiom,
    ! [X: complex,N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N) ).

% norm_power
tff(fact_5284_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5285_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),N) = aa_nat_complex(power_power_complex(Z),N) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5286_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5287_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5288_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_5289_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_5290_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_mult_ineq
tff(fact_5291_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_mult_ineq
tff(fact_5292_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_5293_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_5294_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E)) ) ).

% norm_triangle_lt
tff(fact_5295_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E)) ) ).

% norm_triangle_lt
tff(fact_5296_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C))) ) ).

% norm_add_leD
tff(fact_5297_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C))) ) ).

% norm_add_leD
tff(fact_5298_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E)) ) ).

% norm_triangle_le
tff(fact_5299_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E)) ) ).

% norm_triangle_le
tff(fact_5300_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_5301_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_5302_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_5303_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_5304_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5305_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5306_norm__power__ineq,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),N))) ).

% norm_power_ineq
tff(fact_5307_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N))) ).

% norm_power_ineq
tff(fact_5308_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E)) ) ).

% norm_triangle_le_diff
tff(fact_5309_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E)) ) ).

% norm_triangle_le_diff
tff(fact_5310_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_5311_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_le
tff(fact_5312_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))) ).

% norm_triangle_ineq4
tff(fact_5313_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))) ).

% norm_triangle_ineq4
tff(fact_5314_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))))) ).

% norm_triangle_sub
tff(fact_5315_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))))) ).

% norm_triangle_sub
tff(fact_5316_norm__diff__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))) ).

% norm_diff_ineq
tff(fact_5317_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B)))) ).

% norm_diff_ineq
tff(fact_5318_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5319_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( aa_nat_complex(power_power_complex(W),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5320_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_5321_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D))))) ).

% norm_diff_triangle_ineq
tff(fact_5322_square__norm__one,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_5323_square__norm__one,axiom,
    ! [X: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_5324_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W))))) ) ) ).

% norm_power_diff
tff(fact_5325_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W))))) ) ) ).

% norm_power_diff
tff(fact_5326_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(X,Y)),Z))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X),Z))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5327_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5328_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,ord_max_rat(X,Y)),Z))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5329_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5330_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5331_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ord_max_int(X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5332_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,ord_max_Code_integer(X,Y)),Z))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z)) ) ) ).

% max_less_iff_conj
tff(fact_5333_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A),B))
     => ( ord_ma741700101516333627d_enat(A,B) = B ) ) ).

% max.absorb4
tff(fact_5334_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_5335_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb4
tff(fact_5336_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_5337_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_5338_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb4
tff(fact_5339_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb4
tff(fact_5340_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
     => ( ord_ma741700101516333627d_enat(A,B) = A ) ) ).

% max.absorb3
tff(fact_5341_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_5342_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb3
tff(fact_5343_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_5344_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_5345_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb3
tff(fact_5346_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb3
tff(fact_5347_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_5348_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) ) ).

% pred_lesseq_max
tff(fact_5349_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) ) ).

% succ_greatereq_min
tff(fact_5350_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,T)) ).

% set_vebt'_def
tff(fact_5351_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
               => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))),vEBT_vebt_delete(Summary,H))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_5352_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(Sn)),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(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
tff(fact_5353_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist),vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_5354_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_5355_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(Sn)),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(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
tff(fact_5356_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist),vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_5357_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                 => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))),vEBT_vebt_delete(Summary,H))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_5358_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)) ) ) ) ) ).

% del_in_range
tff(fact_5359_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_5360_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_le211207098394363844omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_af,A)),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_af,B)))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5361_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ag,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_ag,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5362_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_5363_lambda__zero,axiom,
    aTP_Lamm_ai = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_5364_lambda__zero,axiom,
    aTP_Lamm_aj = times_3573771949741848930nteger(zero_z3403309356797280102nteger) ).

% lambda_zero
tff(fact_5365_lambda__zero,axiom,
    aTP_Lamm_ak = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_5366_lambda__zero,axiom,
    aTP_Lamm_al = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_5367_lambda__zero,axiom,
    aTP_Lamm_am = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_5368_lambda__zero,axiom,
    aTP_Lamm_an = times_times_int(zero_zero_int) ).

% lambda_zero
tff(fact_5369_lambda__one,axiom,
    aTP_Lamm_ao = times_times_assn(one_one_assn) ).

% lambda_one
tff(fact_5370_lambda__one,axiom,
    aTP_Lamm_ap = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_5371_lambda__one,axiom,
    aTP_Lamm_aq = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_5372_lambda__one,axiom,
    aTP_Lamm_ar = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_5373_lambda__one,axiom,
    aTP_Lamm_as = times_times_int(one_one_int) ).

% lambda_one
tff(fact_5374_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_at,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_5375_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_au(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_5376_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_av,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_5377_mult__commute__abs,axiom,
    ! [C: int] : aTP_Lamm_aw(C) = times_times_int(C) ).

% mult_commute_abs
tff(fact_5378_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_less_set_complex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_af,A)),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_af,B)))
    <=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
        & ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5379_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,B))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5380_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ag,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_ag,B))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5381_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_5382_numeral__code_I2_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(bit0(N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).

% numeral_code(2)
tff(fact_5383_numeral__code_I2_J,axiom,
    ! [N: num] : numeral_numeral_real(bit0(N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_code(2)
tff(fact_5384_numeral__code_I2_J,axiom,
    ! [N: num] : numeral_numeral_rat(bit0(N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).

% numeral_code(2)
tff(fact_5385_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,bit0(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_code(2)
tff(fact_5386_numeral__code_I2_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,bit0(N)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_code(2)
tff(fact_5387_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_5388_numeral__code_I3_J,axiom,
    ! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).

% numeral_code(3)
tff(fact_5389_numeral__code_I3_J,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_code(3)
tff(fact_5390_numeral__code_I3_J,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).

% numeral_code(3)
tff(fact_5391_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_code(3)
tff(fact_5392_numeral__code_I3_J,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_code(3)
tff(fact_5393_power__numeral__even,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,bit0(W))) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_5394_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] : aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,bit0(W))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_5395_power__numeral__even,axiom,
    ! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,bit0(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_5396_power__numeral__even,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,bit0(W))) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_5397_power__numeral__even,axiom,
    ! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,bit0(W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_5398_power__numeral__even,axiom,
    ! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,bit0(W))) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_even
tff(fact_5399_power__numeral__odd,axiom,
    ! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5400_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] : aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5401_power__numeral__odd,axiom,
    ! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5402_power__numeral__odd,axiom,
    ! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5403_power__numeral__odd,axiom,
    ! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5404_power__numeral__odd,axiom,
    ! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_int_int(times_times_int(aa_int_int(times_times_int(Z),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))) ).

% power_numeral_odd
tff(fact_5405_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax,X)) ).

% set_decode_def
tff(fact_5406_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat)))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_5407_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_5408_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat)))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_5409_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_5410_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_5411_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),ord_ma741700101516333627d_enat(X,Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),X))
        | pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5412_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5413_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),ord_max_rat(X,Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5414_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5415_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5416_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(ord_max_int(X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5417_less__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z),ord_max_Code_integer(X,Y)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z),X))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z),Y)) ) ) ).

% less_max_iff_disj
tff(fact_5418_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(B,C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5419_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5420_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,ord_max_rat(B,C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5421_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5422_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5423_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ord_max_int(B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5424_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,ord_max_Code_integer(B,C)),A))
     => ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
         => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_5425_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
    <=> ( ( A = ord_ma741700101516333627d_enat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5426_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5427_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = ord_max_rat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5428_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5429_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5430_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( ( A = aa_int_int(ord_max_int(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5431_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
    <=> ( ( A = ord_max_Code_integer(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5432_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),ord_ma741700101516333627d_enat(A,B))) ) ).

% max.strict_coboundedI1
tff(fact_5433_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_5434_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),ord_max_rat(A,B))) ) ).

% max.strict_coboundedI1
tff(fact_5435_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_5436_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_5437_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(ord_max_int(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_5438_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.strict_coboundedI1
tff(fact_5439_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),B))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),ord_ma741700101516333627d_enat(A,B))) ) ).

% max.strict_coboundedI2
tff(fact_5440_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_5441_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),ord_max_rat(A,B))) ) ).

% max.strict_coboundedI2
tff(fact_5442_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_5443_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_5444_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(ord_max_int(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_5445_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.strict_coboundedI2
tff(fact_5446_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa4))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa4 = zero_zero_nat )
                 => pp(A4) )
                & ( ( Xa4 != zero_zero_nat )
                 => ( ( ( Xa4 = one_one_nat )
                     => pp(B4) )
                    & ( Xa4 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
             => ~ ( ( Xa4 != Mi2 )
                 => ( ( Xa4 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                               => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_5447_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)))))))))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_5448_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)) ).

% vebt_insert.simps(5)
tff(fact_5449_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa4))
      <=> pp(Y) )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa4 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa4 != zero_zero_nat )
                   => ( ( ( Xa4 = one_one_nat )
                       => pp(B4) )
                      & ( Xa4 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa4 != Mi2 )
                         => ( ( Xa4 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_5450_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa4))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( ( Xa4 = zero_zero_nat )
               => pp(A4) )
              & ( ( Xa4 != zero_zero_nat )
               => ( ( ( Xa4 = one_one_nat )
                   => pp(B4) )
                  & ( Xa4 = one_one_nat ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
         => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
           => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( ( Xa4 != Mi2 )
                     => ( ( Xa4 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_5451_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(X),Xa4) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != aa_num_nat(numeral_numeral_nat,bit0(one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
             => ( Y != aa_num_nat(numeral_numeral_nat,bit0(one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)
               => ( Y != aa_num_nat(numeral_numeral_nat,bit0(one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)))))))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_5452_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa4) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa4 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B4) ) )
                & ( ( Xa4 != zero_zero_nat )
                 => ( ( ( Xa4 = one_one_nat )
                     => ( Y = vEBT_Leaf(A4,fTrue) ) )
                    & ( ( Xa4 != one_one_nat )
                     => ( Y = vEBT_Leaf(A4,B4) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa4),Xa4)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_5453_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = one_one_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = one_one_nat ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_5454_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = some_nat(Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_5455_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = some_nat(Ma) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_5456_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),X) = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),Summary))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_5457_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa4) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( Xa4 = zero_zero_nat )
             => ( Y != vEBT_Leaf(fFalse,B4) ) ) )
       => ( ! [A4: bool] :
              ( ? [B4: bool] : X = vEBT_Leaf(A4,B4)
             => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf(A4,fFalse) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ? [N2: nat] : Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,N2))
                 => ( Y != vEBT_Leaf(A4,B4) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                     => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                             => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                             => ( ( ( ( Xa4 = Mi2 )
                                    & ( Xa4 = Ma2 ) )
                                 => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                                & ( ~ ( ( Xa4 = Mi2 )
                                      & ( Xa4 = Ma2 ) )
                                 => ( Y = 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high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_5458_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(X),Xa4) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
           => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
             => ( ? [N2: nat] : Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                             => ( ( ( ( Xa4 = Mi2 )
                                    & ( Xa4 = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa4 = Mi2 )
                                      & ( Xa4 = Ma2 ) )
                                 => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_5459_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa4) = Y )
     => ( ! [Uu: bool,B4: bool] :
            ( ( X = vEBT_Leaf(Uu,B4) )
           => ( ( Xa4 = zero_zero_nat )
             => ~ ( ( pp(B4)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B4)
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: bool,Uw: bool] : X = vEBT_Leaf(Uv,Uw)
           => ( ? [N2: nat] : Xa4 = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                           => ( Y = some_nat(Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                           => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_5460_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa4) = Y )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: bool] :
              ( ? [Uw: bool] : X = vEBT_Leaf(A4,Uw)
             => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A4)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A4)
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ? [Va: nat] : Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ~ ( ( pp(B4)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B4)
                       => ( ( pp(A4)
                           => ( Y = some_nat(zero_zero_nat) ) )
                          & ( ~ pp(A4)
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(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 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                             => ( Y = some_nat(Ma2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa4),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_5461_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))),Y)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))),Y))),Y))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))),Y)))) ) ) ).

% div_half_nat
tff(fact_5462_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(X),Xa4) = Y )
     => ( ( ? [Uu: bool,B4: bool] : X = vEBT_Leaf(Uu,B4)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: bool,Uw: bool] : X = vEBT_Leaf(Uv,Uw)
           => ( ? [N2: nat] : Xa4 = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                           => ( Y = one_one_nat ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                           => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_5463_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(X),Xa4) = Y )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,Uw: bool] : X = vEBT_Leaf(A4,Uw)
           => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
             => ( ? [Va: nat] : Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,Va))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                             => ( Y = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                             => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_5464_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(X),Xa4) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)
               => ( Y != aa_num_nat(numeral_numeral_nat,bit0(one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat)) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_5465_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_5466_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(X),Xa4) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa4)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),zero_zero_nat)),zero_zero_nat)))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_5467_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,Va2: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf(A,B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_5468_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: bool,Uv2: bool] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf(Uu2,Uv2)),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_5469_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: bool,Uw2: bool,N: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Leaf(Uv2,Uw2)),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_5470_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: bool,B: bool] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Leaf(Uu2,B)),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_5471_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2),Vc)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_5472_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: bool,Uw2: bool] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf(A,Uw2)),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_5473_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_5474_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_5475_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2),Uz2)),X) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_5476_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_5477_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_5478_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% pred_bound_height'
tff(fact_5479_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c2(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% succ'_bound_height
tff(fact_5480_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X) = Y )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: bool] : X = vEBT_Leaf(fTrue,Uv)
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu: bool] : X = vEBT_Leaf(Uu,fTrue)
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2)
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_5481_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)),X) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_5482_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r2(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% member_bound_height'
tff(fact_5483_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d2(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% pred_bound_size_univ'
tff(fact_5484_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c2(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% succ_bound_size_univ'
tff(fact_5485_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),zero_zero_nat)),zero_zero_nat)))))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_5486_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_5487_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_5488_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(X),Xa4) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_5489_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_5490_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(X),Xa4) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
           => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
             => ( ? [N2: nat] : Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,N2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one))))),one_one_nat))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_vebt_delete(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))))))))),one_one_nat)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),one_one_nat)))))),one_one_nat)))))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_5491_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa4))
     => ( ! [Uu: bool,Uv: bool] : X != vEBT_Leaf(Uu,Uv)
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb)
               => ( ( Xa4 = Mi2 )
                  | ( Xa4 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( ( Xa4 = Mi2 )
                    | ( Xa4 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_5492_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa4))
      <=> pp(Y) )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => pp(Y) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb)
               => ( pp(Y)
                <=> ~ ( ( Xa4 = Mi2 )
                      | ( Xa4 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa4 = Mi2 )
                        | ( Xa4 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_5493_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa4))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( ( Xa4 = zero_zero_nat )
               => pp(A4) )
              & ( ( Xa4 != zero_zero_nat )
               => ( ( ( Xa4 = one_one_nat )
                   => pp(B4) )
                  & ( Xa4 = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_5494_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_min_max
tff(fact_5495_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,vEBT_vebt_buildup(N)),X)) ).

% buildup_nothing_in_leaf
tff(fact_5496_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,T),X))
        | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,T),X)) ) ) ).

% both_member_options_def
tff(fact_5497_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,Tree),N))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Tree),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,Tree),X))
          | pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_5498_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,plus_p5714425477246183910nteger(U,one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5499_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: bool,Uv2: bool,Uw2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,vEBT_Leaf(Uu2,Uv2)),Uw2)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_5500_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Ux2)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_5501_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz2)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_5502_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_5503_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_5504_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S)),X) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_5505_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2),Vb2)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_5506_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).

% maxt_bound
tff(fact_5507_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz2)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_5508_mint__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).

% mint_bound
tff(fact_5509_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_5510_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_5511_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S)),X) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_5512_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),X) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_5513_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).

% insert'_bound_height
tff(fact_5514_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B4),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_5515_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X) = Y )
     => ( ! [A4: bool] :
            ( ? [B4: bool] : X = vEBT_Leaf(A4,B4)
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A4),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_5516_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),Vd)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_5517_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeList),S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_5518_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList),Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList)))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_5519_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))))))))),one_one_nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_5520_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))))))),one_one_nat)))) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_5521_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(X),Xa4) = Y )
     => ( ( ? [Uu: bool,Uv: bool] : X = vEBT_Leaf(Uu,Uv)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: bool,Uw: bool] : X = vEBT_Leaf(A4,Uw)
           => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ? [Va: nat] : Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B4),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_5522_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(X),Xa4) = Y )
     => ( ( ? [Uu: bool,B4: bool] : X = vEBT_Leaf(Uu,B4)
         => ( ( Xa4 = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv: bool,Uw: bool] : X = vEBT_Leaf(Uv,Uw)
           => ( ? [N2: nat] : Xa4 = aa_nat_nat(suc,N2)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))))))),one_one_nat)))) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_5523_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa4))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb)
           => ~ ( ( Xa4 = Mi2 )
                | ( Xa4 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
             => ~ ( ( Xa4 = Mi2 )
                  | ( Xa4 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_5524_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),X) = 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ist,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))))))))),one_one_nat)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),one_one_nat)))))),one_one_nat)))))) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_5525_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa4))
      <=> pp(Y) )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa4 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa4 != zero_zero_nat )
                   => ( ( ( Xa4 = one_one_nat )
                       => pp(B4) )
                      & ( Xa4 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)
           => pp(Y) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_5526_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa4))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa4 = zero_zero_nat )
                 => pp(A4) )
                & ( ( Xa4 != zero_zero_nat )
                 => ( ( ( Xa4 = one_one_nat )
                     => pp(B4) )
                    & ( Xa4 = one_one_nat ) ) ) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_5527_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [N2: nat] :
                      ( ( Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xa4)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),Xa4)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xa4)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = 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t,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),one_one_nat)))))),one_one_nat)))))) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_5528_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,B4: bool] :
              ( ( X = vEBT_Leaf(Uu,B4) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf(Uu,B4),zero_zero_nat)) ) ) )
         => ( ! [Uv: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv,Uw) )
               => ! [N2: nat] :
                    ( ( Xa4 = aa_nat_nat(suc,N2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Xa4)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa4)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xa4)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))))))),one_one_nat)))) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_5529_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(A4,Uw) )
               => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf(A4,Uw),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [Va: nat] :
                      ( ( Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B4),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3),Xa4)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Xa4)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xa4)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))))))))),one_one_nat)))) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_5530_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_pred(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ay(T),X)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_5531_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_succ(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_az(T),X)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_5532_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,B4: bool] :
              ( ( X = vEBT_Leaf(Uu,B4) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( ( pp(B4)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B4)
                     => ( Y = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,B4),zero_zero_nat)) ) ) )
         => ( ! [Uv: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv,Uw) )
               => ! [N2: nat] :
                    ( ( Xa4 = aa_nat_nat(suc,N2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Xa4)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa4)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xa4)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                             => ( Y = some_nat(Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_5533_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_5534_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_5535_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_5536_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_5537_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_5538_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_5539_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_5540_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
     => ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_5541_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_5542_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( set_or66887138388493659n_real(A,B) = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
tff(fact_5543_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( set_or3540276404033026485et_nat(A,B) = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_5544_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_5545_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_5546_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_5547_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_5548_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or66887138388493659n_real(A,B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5549_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5550_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5551_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5552_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5553_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5554_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A,B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5555_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5556_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5557_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5558_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A,B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5559_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
    <=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5560_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_5561_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_5562_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(A4,Uw) )
               => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( ( pp(A4)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A4)
                       => ( Y = none_nat ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,Uw),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [Va: nat] :
                      ( ( Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( ( pp(B4)
                           => ( Y = some_nat(one_one_nat) ) )
                          & ( ~ pp(B4)
                           => ( ( pp(A4)
                               => ( Y = some_nat(zero_zero_nat) ) )
                              & ( ~ pp(A4)
                               => ( Y = none_nat ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3),Xa4)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Xa4)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xa4)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                               => ( Y = some_nat(Ma2) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                               => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa4),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))),none_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_5563_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [N2: nat] :
                      ( ( Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xa4)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),Xa4)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xa4)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                               => ( ( ( ( Xa4 = Mi2 )
                                      & ( Xa4 = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa4 = Mi2 )
                                        & ( Xa4 = Ma2 ) )
                                   => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_5564_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = vEBT_Leaf(fFalse,B4) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf(A4,fFalse) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [N2: nat] :
                      ( ( Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                     => ( ( Y = vEBT_Leaf(A4,B4) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xa4)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2),Xa4)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                       => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2),Xa4)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                               => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)) )
                               => ( ( ( ( Xa4 = Mi2 )
                                      & ( Xa4 = Ma2 ) )
                                   => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) )
                                  & ( ~ ( ( Xa4 = Mi2 )
                                        & ( Xa4 = Ma2 ) )
                                   => ( Y = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_time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(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),vEBT_vebt_delete(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fimplies(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(fimplies(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2))))))),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_5565_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,B4: bool] :
              ( ( X = vEBT_Leaf(Uu,B4) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,B4),zero_zero_nat)) ) ) )
         => ( ! [Uv: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv,Uw) )
               => ! [N2: nat] :
                    ( ( Xa4 = aa_nat_nat(suc,N2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Xa4)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2),Xa4)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xa4)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                             => ( Y = one_one_nat ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                             => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_5566_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xa4)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xa4)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa4)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_5567_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( ( Xa4 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B4) ) )
                  & ( ( Xa4 != zero_zero_nat )
                   => ( ( ( Xa4 = one_one_nat )
                       => ( Y = vEBT_Leaf(A4,fTrue) ) )
                      & ( ( Xa4 != one_one_nat )
                       => ( Y = vEBT_Leaf(A4,B4) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xa4)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xa4)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa4),Xa4)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa4)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Xa4),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),Summary2))),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_5568_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ( Xa4 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),zero_zero_nat)) ) ) )
         => ( ! [A4: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(A4,Uw) )
               => ( ( Xa4 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,Uw),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [Va: nat] :
                      ( ( Xa4 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va3),Xa4)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Xa4)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xa4)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                               => ( Y = one_one_nat ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_5569_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xa4)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xa4)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary2),Xa4)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),Mi2),Xa4)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),one_one_nat))),one_one_nat) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_5570_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xa4)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xa4)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2),Xa4)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_nat)))))))))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_5571_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(X),Xa4) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xa4)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xa4)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2),Xa4)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa4),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa4)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),zero_zero_nat)),zero_zero_nat)))))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_5572_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_real,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_real(TreeList4) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),X)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X)))) ) ) ).

% tcd
tff(fact_5573_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_bool,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_bool(TreeList4) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),X)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X)))) ) ) ).

% tcd
tff(fact_5574_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_nat,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_nat(TreeList4) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),X)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X)))) ) ) ).

% tcd
tff(fact_5575_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_int,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_int(TreeList4) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),X)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y)),list_u6098035379799741383_VEBTi(Tree_is,I,X)))) ) ) ).

% tcd
tff(fact_5576_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList)))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% txe
tff(fact_5577_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(T),insert_nat(X,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_5578_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = minus_minus_set_nat(vEBT_set_vebt(T),insert_nat(X,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_5579_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList)))
     => entails(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y)))) ) ).

% local.ext
tff(fact_5580_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_5581_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% recomp
tff(fact_5582_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList)))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),Rest)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% repack
tff(fact_5583_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_5584_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_5585_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5586_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5587_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_5588_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),insert_real(X,set_real2(Xs)))) ).

% set_update_subset_insert
tff(fact_5589_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),insert_int(X,set_int2(Xs)))) ).

% set_update_subset_insert
tff(fact_5590_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_5591_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,X)),insert_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_5592_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X))),insert_nat(X,set_nat2(Xs)))) ).

% set_update_subset_insert
tff(fact_5593_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: real,B: real] : set_or66887138388493659n_real(A,B) = minus_minus_set_real(set_or1222579329274155063t_real(A,B),insert_real(B,bot_bot_set_real)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5594_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: nat,B: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = minus_minus_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B),insert_nat(B,bot_bot_set_nat)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5595_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: int,B: int] : set_or4662586982721622107an_int(A,B) = minus_minus_set_int(set_or1266510415728281911st_int(A,B),insert_int(B,bot_bot_set_int)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5596_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: code_integer,B: code_integer] : set_or8404916559141939852nteger(A,B) = minus_2355218937544613996nteger(set_or189985376899183464nteger(A,B),insert_Code_integer(B,bot_bo3990330152332043303nteger)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5597_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_5598_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
     => ( insert_VEBT_VEBT(nth_VEBT_VEBT(L,I),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,X))) = insert_VEBT_VEBT(X,set_VEBT_VEBT2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5599_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)))
     => ( insert_VEBT_VEBTi(nth_VEBT_VEBTi(L,I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,X))) = insert_VEBT_VEBTi(X,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5600_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(L)))
     => ( insert_real(aa_nat_real(nth_real(L),I),set_real2(list_update_real(L,I,X))) = insert_real(X,set_real2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5601_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( insert_bool(aa_nat_bool(nth_bool(L),I),set_bool2(list_update_bool(L,I,X))) = insert_bool(X,set_bool2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5602_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( insert_nat(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,X))) = insert_nat(X,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5603_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( insert_int(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,X))) = insert_int(X,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5604_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5605_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5606_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE123837653852366616t_assn,Xsi: list_int] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8648204552663881920BT_int(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5607_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5608_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5609_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE1225457145571040039t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2809031099982602151Ti_nat(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5610_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5641251438918192003t_assn,Xsi: list_int] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2806540629473551875Ti_int(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB9043990991474615620t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5611_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5612_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5613_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L234762979517870878al_nat(insert_nat(I,I5),A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5614_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(insert_nat(I,I5),A2,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X1),X22)),vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5615_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(insert_nat(I,I5),A2,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,X1),X22)),vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5616_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(insert_nat(I,I5),A2,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,X1),X22)),vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5617_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(insert_nat(I,I5),A2,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,X1),X22)),vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5618_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(insert_nat(I,I5),A2,list_update_real(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,X1),X22)),vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5619_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo5394993412183903963T_assn,X1: bool,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( vEBT_L2992182275864018813T_VEBT(insert_nat(I,I5),A2,list_update_bool(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A2,X1),X22)),vEBT_L2992182275864018813T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5620_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_bool,A2: fun_bo7535889243153246212i_assn,X1: bool,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( vEBT_L6372884366951783724_VEBTi(insert_nat(I,I5),A2,list_update_bool(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A2,X1),X22)),vEBT_L6372884366951783724_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5621_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L8511957252848910786T_VEBT(insert_nat(I,I5),A2,list_update_nat(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A2,X1),X22)),vEBT_L8511957252848910786T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5622_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L7489483478785760935_VEBTi(insert_nat(I,I5),A2,list_update_nat(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A2,X1),X22)),vEBT_L7489483478785760935_VEBTi(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5623_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A2: fun_in7755481285845205148T_assn,X1: int,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( vEBT_L2018189785592951398T_VEBT(insert_nat(I,I5),A2,list_update_int(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A2,X1),X22)),vEBT_L2018189785592951398T_VEBT(I5,A2,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5624_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,Z)))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5625_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5626_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5627_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: fun_VE123837653852366616t_assn,Xsi: list_int] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5628_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5577048379444288083T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5629_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE2561185598974572172i_assn,Xsi: list_VEBT_VEBTi] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5630_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE1225457145571040039t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5631_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: fun_VE5641251438918192003t_assn,Xsi: list_int] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB9043990991474615620t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5632_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5633_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5634_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(I),I5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5635_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5636_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5637_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5638_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5639_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5640_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5641_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB9043990991474615620t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5642_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5643_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5644_listI__assn__reinsert,axiom,
    ! [P: assn,A2: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi)),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5645_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,X: vEBT_VEBT,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X),Xi2)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I5,A2,list_u1324408373059187874T_VEBT(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5646_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,X: vEBT_VEBTi,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,X),Xi2)),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2497118539674116125T_VEBT(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5647_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,X: vEBT_VEBTi,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,X),Xi2)),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L886525131989349516_VEBTi(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5648_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,X: real,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,X),Xi2)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I5,A2,list_update_real(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5649_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,X: real,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,X),Xi2)),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L7851252805511451907_VEBTi(I5,A2,list_update_real(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5650_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_bo5394993412183903963T_assn,X: bool,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A2,X),Xi2)),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2992182275864018813T_VEBT(I5,A2,list_update_bool(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5651_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_bo7535889243153246212i_assn,X: bool,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A2,X),Xi2)),vEBT_L6372884366951783724_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L6372884366951783724_VEBTi(I5,A2,list_update_bool(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5652_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_na4489259242115493880T_assn,X: nat,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A2,X),Xi2)),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L8511957252848910786T_VEBT(I5,A2,list_update_nat(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5653_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_na6467670216208870567i_assn,X: nat,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A2,X),Xi2)),vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L7489483478785760935_VEBTi(I5,A2,list_update_nat(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5654_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: fun_in7755481285845205148T_assn,X: int,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_int,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A2,X),Xi2)),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2018189785592951398T_VEBT(I5,A2,list_update_int(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5655_big__assn__simp,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,L: nat,X: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList)))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),X)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xaa),vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xaa),vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))),list_u6098035379799741383_VEBTi(Tree_is,H,X)))) ) ).

% big_assn_simp
tff(fact_5656_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xb3: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList)))
     => ( ( Xaa = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),X)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xb3),vEBT_vebt_mint(Xaa))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,X))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Xb3),vEBT_vebt_mint(Xaa))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,Xaa)),list_u6098035379799741383_VEBTi(Tree_is,H,X)))) ) ) ).

% big_assn_simp'
tff(fact_5657_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa4))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4))
               => ( ( ( Xa4 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa4 != zero_zero_nat )
                   => ( ( ( Xa4 = one_one_nat )
                       => pp(B4) )
                      & ( Xa4 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xa4)) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xa4)) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2),Xa4)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4))
                       => ( ( Xa4 != Mi2 )
                         => ( ( Xa4 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_5658_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa4))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa4 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa4 != zero_zero_nat )
                     => ( ( ( Xa4 = one_one_nat )
                         => pp(B4) )
                        & ( Xa4 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xa4)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xa4)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2),Xa4)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa4 != Mi2 )
                           => ( ( Xa4 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_5659_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa4))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4))
               => ( ( ( Xa4 = zero_zero_nat )
                   => pp(A4) )
                  & ( ( Xa4 != zero_zero_nat )
                   => ( ( ( Xa4 = one_one_nat )
                       => pp(B4) )
                      & ( Xa4 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw),Xa4)) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),Xa4))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_5660_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa4))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4))
               => ~ ( ( ( Xa4 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa4 != zero_zero_nat )
                     => ( ( ( Xa4 = one_one_nat )
                         => pp(B4) )
                        & ( Xa4 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),Xa4))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_5661_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(plus_plus_int(one_one_int),N)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(plus_plus_int(one_one_int),N)) = insert_int(aa_int_int(plus_plus_int(one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5662_simp__from__to,axiom,
    ! [J2: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I))
       => ( set_or1266510415728281911st_int(I,J2) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I))
       => ( set_or1266510415728281911st_int(I,J2) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(plus_plus_int(I),one_one_int),J2)) ) ) ) ).

% simp_from_to
tff(fact_5663_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,X),Xa4))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4))
               => ~ ( ( ( Xa4 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa4 != zero_zero_nat )
                     => ( ( ( Xa4 = one_one_nat )
                         => pp(B4) )
                        & ( Xa4 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xa4))
                 => ~ ( ( Xa4 != Mi2 )
                     => ( ( Xa4 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa4))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_5664_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,X),Xa4))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa4 = zero_zero_nat )
                     => pp(A4) )
                    & ( ( Xa4 != zero_zero_nat )
                     => ( ( ( Xa4 = one_one_nat )
                         => pp(B4) )
                        & ( Xa4 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa4)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw),Xa4)) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),Xa4)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_5665_IH,axiom,
    ! [Xa4: nat,Xb3: nat,Ti: vEBT_VEBTi] :
      ( ( xa != mi )
     => ( ( xa != ma )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi))
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),xa))
           => ( ( Xa4 = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,bit0(one)))) )
             => ( ( Xb3 = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,bit0(one)))) )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa4),size_s6755466524823107622T_VEBT(treeList)))
                 => hoare_2774773592644289642e_bool(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(treeList,Xa4)),Ti),vEBT_V854960066525838166emberi(nth_VEBT_VEBT(treeList,Xa4),Ti,Xb3),aa_VEB1972364096475303967l_assn(aa_nat617759954745536040l_assn(aTP_Lamm_ba(Xa4),Xb3),Ti)) ) ) ) ) ) ) ) ).

% IH
tff(fact_5666_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat,Y: bool] :
      ( ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa4))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa4)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xa4)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb) )
                 => ( ( pp(Y)
                    <=> ( ( Xa4 = Mi2 )
                        | ( Xa4 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb),Xa4)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa4 = Mi2 )
                          | ( Xa4 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa4)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                             => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa4)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_5667_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( ~ pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa4))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Uu: bool,Uv: bool] :
              ( ( X = vEBT_Leaf(Uu,Uv) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu,Uv),Xa4)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xa4)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb),Xa4))
                   => ( ( Xa4 = Mi2 )
                      | ( Xa4 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa4))
                     => ( ( Xa4 = Mi2 )
                        | ( Xa4 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa4))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_5668_ent__pure__pre__iff,axiom,
    ! [P: assn,B: bool,Q: assn] :
      ( entails(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),Q)
    <=> ( pp(B)
       => entails(P,Q) ) ) ).

% ent_pure_pre_iff
tff(fact_5669_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa4: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,X),Xa4))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa4))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb),Xa4))
               => ~ ( ( Xa4 = Mi2 )
                    | ( Xa4 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2),Xa4))
                 => ~ ( ( Xa4 = Mi2 )
                      | ( Xa4 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2),Xa4))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,bit0(one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_5670_heaphelp,axiom,
    ! [Xa4: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb3: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xa4,Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb3))),aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,none_nat),none_nat)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),N))))),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Xc),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,N,Xa4),Xb3)))),H)
     => rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,N,TreeList),Summary)),Xc),H) ) ).

% heaphelp
tff(fact_5671_heaphelp,axiom,
    ! [Xa4: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb3: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xa4,Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb3))),aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,none_P5556105721700978146at_nat),none_P5556105721700978146at_nat)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),N))))),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Xc),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,N,Xa4),Xb3)))),H)
     => rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,N,TreeList),Summary)),Xc),H) ) ).

% heaphelp
tff(fact_5672_heaphelp,axiom,
    ! [Xa4: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb3: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xa4,Tree_is)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList),Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb3))),aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_option_num_bool(aa_opt4409681509335781342m_bool(fequal_option_num,none_num),none_num)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),N))))),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Xc),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,N,Xa4),Xb3)))),H)
     => rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,N,TreeList),Summary)),Xc),H) ) ).

% heaphelp
tff(fact_5673_merge__pure__star,axiom,
    ! [A: bool,B: bool] : aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,A)),aa_bool_assn(pure_assn,B)) = aa_bool_assn(pure_assn,aa_bool_bool(fconj(A),B)) ).

% merge_pure_star
tff(fact_5674_ent__pure__pre__iff__sng,axiom,
    ! [B: bool,Q: assn] :
      ( entails(aa_bool_assn(pure_assn,B),Q)
    <=> ( pp(B)
       => entails(one_one_assn,Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_5675_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( rep_assn(aa_assn_assn(times_times_assn(P),Q),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ( rep_assn(P,produc7507926704131184380et_nat(H,bot_bot_set_nat))
        & rep_assn(Q,produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_5676_mod__pure__star__dist,axiom,
    ! [P: assn,B: bool,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),H)
    <=> ( rep_assn(P,H)
        & pp(B) ) ) ).

% mod_pure_star_dist
tff(fact_5677_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: bool] :
      ( entails(P,aa_bool_assn(pure_assn,B))
    <=> ( ! [H2: produc3658429121746597890et_nat] :
            ( rep_assn(P,H2)
           => pp(B) )
        & entails(P,one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_5678_mod__h__bot__iff_I4_J,axiom,
    ! [Q2: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : ~ rep_assn(snga_assn_VEBT_VEBTi(Q2,Y),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_h_bot_iff(4)
tff(fact_5679_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: bool] :
      ( entails(P,aa_assn_assn(times_times_assn(Q),aa_bool_assn(pure_assn,B)))
    <=> ( ! [H2: produc3658429121746597890et_nat] :
            ( rep_assn(P,H2)
           => pp(B) )
        & entails(P,Q) ) ) ).

% ent_pure_post_iff
tff(fact_5680_mod__starD,axiom,
    ! [A2: assn,B2: assn,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(A2),B2),H)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( rep_assn(A2,H1)
          & rep_assn(B2,H22) ) ) ).

% mod_starD
tff(fact_5681_mod__starE,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(A),B),H)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : rep_assn(A,X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ rep_assn(B,H_2) ) ) ).

% mod_starE
tff(fact_5682_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
    <=> ! [H2: produc3658429121746597890et_nat] :
          ( rep_assn(P,H2)
         => rep_assn(Q,H2) ) ) ).

% entails_def
tff(fact_5683_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( rep_assn(P,H3)
         => rep_assn(Q,H3) )
     => entails(P,Q) ) ).

% entailsI
tff(fact_5684_entailsD,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( entails(P,Q)
     => ( rep_assn(P,H)
       => rep_assn(Q,H) ) ) ).

% entailsD
tff(fact_5685_ent__fwd,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( rep_assn(P,H)
     => ( entails(P,Q)
       => rep_assn(Q,H) ) ) ).

% ent_fwd
tff(fact_5686_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L1930518968523514909l_real(A2,Xs,Xsi),H)
     => ( size_size_list_real(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5687_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_bo8885878543415060347l_assn,Xs: list_bool,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L5107238702033023424l_real(A2,Xs,Xsi),H)
     => ( size_size_list_real(Xsi) = size_size_list_bool(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5688_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_na448437756743965086l_assn,Xs: list_nat,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L6102073776069194049t_real(A2,Xs,Xsi),H)
     => ( size_size_list_real(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5689_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_in6446958447386763770l_assn,Xs: list_int,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L8288995350762215837t_real(A2,Xs,Xsi),H)
     => ( size_size_list_real(Xsi) = size_size_list_int(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5690_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re2206215847901373695l_assn,Xs: list_real,Xsi: list_bool,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L870199270341830246l_bool(A2,Xs,Xsi),H)
     => ( size_size_list_bool(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5691_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_bo8172996160782635326l_assn,Xs: list_bool,Xsi: list_bool,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L6562251693209944067l_bool(A2,Xs,Xsi),H)
     => ( size_size_list_bool(Xsi) = size_size_list_bool(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5692_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_na5136419689362302555l_assn,Xs: list_nat,Xsi: list_bool,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L2559740421116301378t_bool(A2,Xs,Xsi),H)
     => ( size_size_list_bool(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5693_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_in4945713774130038527l_assn,Xs: list_int,Xsi: list_bool,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L8658821491502452966t_bool(A2,Xs,Xsi),H)
     => ( size_size_list_bool(Xsi) = size_size_list_int(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5694_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L1446010312343316929al_nat(A2,Xs,Xsi),H)
     => ( size_size_list_nat(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5695_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_bo8438563643002758303t_assn,Xs: list_bool,Xsi: list_nat,H: produc3658429121746597890et_nat] :
      ( rep_assn(vEBT_L7520539408223738084ol_nat(A2,Xs,Xsi),H)
     => ( size_size_list_nat(Xsi) = size_size_list_bool(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5696_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(one_one_assn),P) = P ).

% assn_one_left
tff(fact_5697_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(times_times_assn(P),Q) = aa_assn_assn(times_times_assn(Q),P) ).

% assn_times_comm
tff(fact_5698_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R3: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),R3) = aa_assn_assn(times_times_assn(P),aa_assn_assn(times_times_assn(Q),R3)) ).

% assn_times_assoc
tff(fact_5699_ent__iffI,axiom,
    ! [A2: assn,B2: assn] :
      ( entails(A2,B2)
     => ( entails(B2,A2)
       => ( A2 = B2 ) ) ) ).

% ent_iffI
tff(fact_5700_ent__refl,axiom,
    ! [P: assn] : entails(P,P) ).

% ent_refl
tff(fact_5701_ent__trans,axiom,
    ! [P: assn,Q: assn,R3: assn] :
      ( entails(P,Q)
     => ( entails(Q,R3)
       => entails(P,R3) ) ) ).

% ent_trans
tff(fact_5702_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q5: assn] :
      ( entails(P,P2)
     => ( entails(Q,Q5)
       => entails(aa_assn_assn(times_times_assn(P),Q),aa_assn_assn(times_times_assn(P2),Q5)) ) ) ).

% ent_star_mono
tff(fact_5703_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5704_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5705_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L8648204552663881920BT_int(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5706_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L2497118539674116125T_VEBT(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5707_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L886525131989349516_VEBTi(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5708_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L2809031099982602151Ti_nat(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5709_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB9043990991474615620t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L2806540629473551875Ti_int(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5710_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5711_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L7851252805511451907_VEBTi(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5712_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I5,A2,Xs,Xsi)),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5713_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,X: vEBT_VEBT,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X),Xi2)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I5,A2,list_u1324408373059187874T_VEBT(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5714_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,X: vEBT_VEBTi,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,X),Xi2)),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L2497118539674116125T_VEBT(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5715_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,X: vEBT_VEBTi,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,X),Xi2)),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L886525131989349516_VEBTi(I5,A2,list_u6098035379799741383_VEBTi(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5716_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,X: real,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,X),Xi2)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I5,A2,list_update_real(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5717_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,X: real,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,X),Xi2)),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L7851252805511451907_VEBTi(I5,A2,list_update_real(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5718_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_bo5394993412183903963T_assn,X: bool,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_boo5145214906438187220T_assn(A2,X),Xi2)),vEBT_L2992182275864018813T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L2992182275864018813T_VEBT(I5,A2,list_update_bool(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5719_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_bo7535889243153246212i_assn,X: bool,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_bool,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_boo89690317562068533i_assn(A2,X),Xi2)),vEBT_L6372884366951783724_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L6372884366951783724_VEBTi(I5,A2,list_update_bool(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5720_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_na4489259242115493880T_assn,X: nat,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A2,X),Xi2)),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L8511957252848910786T_VEBT(I5,A2,list_update_nat(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5721_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_na6467670216208870567i_assn,X: nat,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A2,X),Xi2)),vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L7489483478785760935_VEBTi(I5,A2,list_update_nat(Xs,I,X),list_u6098035379799741383_VEBTi(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5722_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: fun_in7755481285845205148T_assn,X: int,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_int,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A2,X),Xi2)),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(I5,insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(vEBT_L2018189785592951398T_VEBT(I5,A2,list_update_int(Xs,I,X),list_u1324408373059187874T_VEBT(Xsi,I,Xi2))),F2),C,Q)
           => hoare_2774773592644289642e_bool(P,C,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5723_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L1279224858307276611T_VEBT(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,nth_VEBT_VEBT(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7769146593908314002l_assn(aTP_Lamm_bb(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5724_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(A2,Xs),Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis5809874432478338334l_assn(aTP_Lamm_bc(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5725_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE123837653852366616t_assn,Xs: list_VEBT_VEBT,Xsi: list_int,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L8294436054247626077BT_int(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB392321844297299145t_assn(A2,nth_VEBT_VEBT(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8648204552663881920BT_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis4088241294491955066l_assn(aTP_Lamm_bd(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5726_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L7265847600308530106T_VEBT(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis183111350483865165l_assn(aTP_Lamm_be(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5727_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L1891944875198410415_VEBTi(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A2,nth_VEBT_VEBTi(Xs,I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis4074100605738197896l_assn(aTP_Lamm_bf(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5728_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,Xsi: list_nat,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L8930081998596925642Ti_nat(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8068455526268438248t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis3503319847846646691l_assn(aTP_Lamm_bg(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5729_rule__at__index,axiom,
    ! [P: assn,A2: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,Xsi: list_int,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L8927591528087875366Ti_int(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB9043990991474615620t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_int_assn(aa_VEB9043990991474615620t_assn(A2,nth_VEBT_VEBTi(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis1781686709860263423l_assn(aTP_Lamm_bh(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5730_rule__at__index,axiom,
    ! [P: assn,A2: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L4595930785310033027T_VEBT(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBT(Xsi,I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7705788952316199574l_assn(aTP_Lamm_bi(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5731_rule__at__index,axiom,
    ! [P: assn,A2: fun_re560840401387337987i_assn,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L9060850011106065574_VEBTi(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A2,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis3027762689479947775l_assn(aTP_Lamm_bj(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5732_rule__at__index,axiom,
    ! [P: assn,A2: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,F2: assn,I: nat,C: heap_Time_Heap_bool,Q5: fun_bool_assn,F4: fun_bool_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L1446010312343316929al_nat(A2,Xs,Xsi)),F2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_real(Xs)))
       => ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),F2),C,Q5)
         => ( ! [R: bool] : entails(aa_bool_assn(Q5,R),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A2,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A2,Xs,Xsi))),aa_bool_assn(F4,R)))
           => hoare_2774773592644289642e_bool(P,C,aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis4366942040065530138l_assn(aTP_Lamm_bk(A2),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5733_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_2774773592644289642e_bool(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_5734_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_7629718768684598413on_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_5735_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_3067605981109127869le_nat(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_5736_norm__pre__pure__iff,axiom,
    ! [P: assn,B: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q)
    <=> ( pp(B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_5737_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I3))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D)),D))) ) ).

% foldr_zero
tff(fact_5738_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X2: real,Y3: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => ( pp(aa_set_real_bool(member_real(Y3),set_real2(Xs)))
           => ( X2 = Y3 ) ) )
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
           => ( X2 = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_5739_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),aa_int_rat(ring_1_of_int_rat,Y)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_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
tff(fact_5740_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,Y)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_5741_foldr0,axiom,
    ! [Xs: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C) ).

% foldr0
tff(fact_5742_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D))) ).

% foldr_one
tff(fact_5743_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X2: nat,Y3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( X2 = Y3 ) ) )
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
           => ( X2 = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_5744_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Ys),I3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D))) ) ) ) ).

% foldr_mono
tff(fact_5745_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5746_round__0,axiom,
    archim7778729529865785530nd_rat(zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5747_round__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5748_round__numeral,axiom,
    ! [N: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5749_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_bl,L),zero_zero_nat) = size_size_list_real(L) ).

% foldr_length
tff(fact_5750_foldr__length,axiom,
    ! [L: list_bool] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_bm,L),zero_zero_nat) = size_size_list_bool(L) ).

% foldr_length
tff(fact_5751_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_bn,L),zero_zero_nat) = size_size_list_nat(L) ).

% foldr_length
tff(fact_5752_foldr__length,axiom,
    ! [L: list_int] : aa_nat_nat(foldr_int_nat(aTP_Lamm_bo,L),zero_zero_nat) = size_size_list_int(L) ).

% foldr_length
tff(fact_5753_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_minti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bp(T),Ti)) ).

% vebt_minti_h
tff(fact_5754_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_maxti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bq(T),Ti)) ).

% vebt_maxti_h
tff(fact_5755_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2)))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_minti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_br(I),Ts2),Tsi)) ) ).

% vebt_mintilist
tff(fact_5756_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2)))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_maxti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bs(I),Ts2),Tsi)) ) ).

% vebt_maxtilist
tff(fact_5757_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_bool,K: list_bool,F: fun_bool_fun_nat_nat,G: fun_bool_fun_nat_nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A4: nat,X2: bool] :
              ( member_bool(X2,set_bool2(L))
             => ( aa_nat_nat(aa_bool_fun_nat_nat(F,X2),A4) = aa_nat_nat(aa_bool_fun_nat_nat(G,X2),A4) ) )
         => ( aa_nat_nat(foldr_bool_nat(F,L),A) = aa_nat_nat(foldr_bool_nat(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_5758_foldr__cong,axiom,
    ! [A: real,B: real,L: list_real,K: list_real,F: fun_re7681423839302404990l_real,G: fun_re7681423839302404990l_real] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A4: real,X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),set_real2(L)))
             => ( aa_real_real(aa_rea7340605660300375871l_real(F,X2),A4) = aa_real_real(aa_rea7340605660300375871l_real(G,X2),A4) ) )
         => ( aa_real_real(foldr_real_real(F,L),A) = aa_real_real(foldr_real_real(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_5759_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_nat,K: list_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A4: nat,X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(L)))
             => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X2),A4) = aa_nat_nat(aa_nat_fun_nat_nat(G,X2),A4) ) )
         => ( aa_nat_nat(foldr_nat_nat(F,L),A) = aa_nat_nat(foldr_nat_nat(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_5760_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_bl,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_real(L)) ).

% foldr_length_aux
tff(fact_5761_foldr__length__aux,axiom,
    ! [L: list_bool,A: nat] : aa_nat_nat(foldr_bool_nat(aTP_Lamm_bm,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_bool(L)) ).

% foldr_length_aux
tff(fact_5762_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_bn,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_nat(L)) ).

% foldr_length_aux
tff(fact_5763_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] : aa_nat_nat(foldr_int_nat(aTP_Lamm_bo,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_int(L)) ).

% foldr_length_aux
tff(fact_5764_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_bool_assn,Q5: fun_bool_assn,C: heap_Time_Heap_bool] :
      ( entails(P,P2)
     => ( ! [X2: bool] : entails(aa_bool_assn(Q,X2),aa_bool_assn(Q5,X2))
       => ( hoare_2774773592644289642e_bool(P2,C,Q)
         => hoare_2774773592644289642e_bool(P,C,Q5) ) ) ) ).

% cons_rule
tff(fact_5765_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_option_nat_assn,Q5: fun_option_nat_assn,C: heap_T2636463487746394924on_nat] :
      ( entails(P,P2)
     => ( ! [X2: option_nat] : entails(aa_option_nat_assn(Q,X2),aa_option_nat_assn(Q5,X2))
       => ( hoare_7629718768684598413on_nat(P2,C,Q)
         => hoare_7629718768684598413on_nat(P,C,Q5) ) ) ) ).

% cons_rule
tff(fact_5766_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_nat_assn,Q5: fun_nat_assn,C: heap_Time_Heap_nat] :
      ( entails(P,P2)
     => ( ! [X2: nat] : entails(aa_nat_assn(Q,X2),aa_nat_assn(Q5,X2))
       => ( hoare_3067605981109127869le_nat(P2,C,Q)
         => hoare_3067605981109127869le_nat(P,C,Q5) ) ) ) ).

% cons_rule
tff(fact_5767_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_VEBT_VEBTi_assn,Q5: fun_VEBT_VEBTi_assn,C: heap_T8145700208782473153_VEBTi] :
      ( entails(P,P2)
     => ( ! [X2: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q,X2),aa_VEBT_VEBTi_assn(Q5,X2))
       => ( hoare_1429296392585015714_VEBTi(P2,C,Q)
         => hoare_1429296392585015714_VEBTi(P,C,Q5) ) ) ) ).

% cons_rule
tff(fact_5768_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,Q5: fun_bool_assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => ( ! [X2: bool] : entails(aa_bool_assn(Q,X2),aa_bool_assn(Q5,X2))
       => hoare_2774773592644289642e_bool(P,C,Q5) ) ) ).

% cons_post_rule
tff(fact_5769_cons__post__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Q5: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( ! [X2: option_nat] : entails(aa_option_nat_assn(Q,X2),aa_option_nat_assn(Q5,X2))
       => hoare_7629718768684598413on_nat(P,C,Q5) ) ) ).

% cons_post_rule
tff(fact_5770_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Q5: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( ! [X2: nat] : entails(aa_nat_assn(Q,X2),aa_nat_assn(Q5,X2))
       => hoare_3067605981109127869le_nat(P,C,Q5) ) ) ).

% cons_post_rule
tff(fact_5771_cons__post__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Q5: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( ! [X2: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q,X2),aa_VEBT_VEBTi_assn(Q5,X2))
       => hoare_1429296392585015714_VEBTi(P,C,Q5) ) ) ).

% cons_post_rule
tff(fact_5772_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,R3: assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(P),R3),C,aa_ass2376637134808068204l_assn(aTP_Lamm_bt(Q),R3)) ) ).

% frame_rule
tff(fact_5773_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R3: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),R3),C,aa_ass4112278883933122603t_assn(aTP_Lamm_bu(Q),R3)) ) ).

% frame_rule
tff(fact_5774_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R3: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),R3),C,aa_assn_fun_nat_assn(aTP_Lamm_bv(Q),R3)) ) ).

% frame_rule
tff(fact_5775_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R3: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),R3),C,aa_ass4389913640333755968i_assn(aTP_Lamm_bw(Q),R3)) ) ).

% frame_rule
tff(fact_5776_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( entails(P,P2)
     => ( hoare_2774773592644289642e_bool(P2,C,Q)
       => hoare_2774773592644289642e_bool(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_5777_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( entails(P,P2)
     => ( hoare_7629718768684598413on_nat(P2,C,Q)
       => hoare_7629718768684598413on_nat(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_5778_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( entails(P,P2)
     => ( hoare_3067605981109127869le_nat(P2,C,Q)
       => hoare_3067605981109127869le_nat(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_5779_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( entails(P,P2)
     => ( hoare_1429296392585015714_VEBTi(P2,C,Q)
       => hoare_1429296392585015714_VEBTi(P,C,Q) ) ) ).

% cons_pre_rule
tff(fact_5780_of__int__round__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))),plus_plus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one)))))) ).

% of_int_round_le
tff(fact_5781_of__int__round__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))))) ).

% of_int_round_le
tff(fact_5782_of__int__round__ge,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)))) ).

% of_int_round_ge
tff(fact_5783_of__int__round__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_ge
tff(fact_5784_of__int__round__gt,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).

% of_int_round_gt
tff(fact_5785_of__int__round__gt,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)))) ).

% of_int_round_gt
tff(fact_5786_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_bool,Q: fun_bool_assn] :
      ( ( pp(B)
       => hoare_2774773592644289642e_bool(P,F,Q) )
     => hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_5787_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( pp(B)
       => hoare_7629718768684598413on_nat(P,F,Q) )
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_5788_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( pp(B)
       => hoare_3067605981109127869le_nat(P,F,Q) )
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_5789_norm__pre__pure__rule1,axiom,
    ! [B: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( pp(B)
       => hoare_1429296392585015714_VEBTi(P,F,Q) )
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_5790_lowi__h,axiom,
    ! [X: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(X,N),aa_nat_fun_nat_assn(aTP_Lamm_bx(X),N)) ).

% lowi_h
tff(fact_5791_highi__h,axiom,
    ! [X: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(X,N),aa_nat_fun_nat_assn(aTP_Lamm_by(X),N)) ).

% highi_h
tff(fact_5792_builupicorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))) ).

% builupicorr
tff(fact_5793_builupi_Hcorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))) ).

% builupi'corr
tff(fact_5794_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5795_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F: fun_real_real,Bound: real,I: real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5796_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_bool,F: fun_bool_real,Bound: real,I: real] :
      ( ! [X2: bool] :
          ( member_bool(X2,set_bool2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_bool_real(F,X2)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_bool_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_bool(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5797_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Bound: real,I: real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X2)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_nat(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5798_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_int,F: fun_int_real,Bound: real,I: real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),Bound)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_int(Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5799_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,C: real,G: fun_VEBT_VEBT_real,D: real] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_VEBT_VEBT_real(G,X2)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_5800_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,C: real,G: fun_real_real,D: real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(G,X2)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_5801_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,C: real,G: fun_nat_real,D: real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_nat_real(G,X2)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_5802_f__g__map__foldr__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,C: real,G: fun_int_real,D: real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_int_real(G,X2)))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(G,Xs)),zero_zero_real))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_5803_map__ident,axiom,
    ! [X3: list_nat] : map_nat_nat(aTP_Lamm_ar,X3) = X3 ).

% map_ident
tff(fact_5804_listsum__bound,axiom,
    ! [Xs: list_complex,F: fun_complex_real,Y: real] :
      ( ! [X2: complex] :
          ( member_complex(X2,set_complex2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_complex_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_5805_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Y: real] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_5806_listsum__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,Y: real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_5807_listsum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Y: real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_5808_listsum__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,Y: real] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),Y))) ) ).

% listsum_bound
tff(fact_5809_length__map,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : size_size_list_real(map_VEBT_VEBT_real(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5810_length__map,axiom,
    ! [F: fun_real_real,Xs: list_real] : size_size_list_real(map_real_real(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5811_length__map,axiom,
    ! [F: fun_bool_real,Xs: list_bool] : size_size_list_real(map_bool_real(F,Xs)) = size_size_list_bool(Xs) ).

% length_map
tff(fact_5812_length__map,axiom,
    ! [F: fun_nat_real,Xs: list_nat] : size_size_list_real(map_nat_real(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5813_length__map,axiom,
    ! [F: fun_int_real,Xs: list_int] : size_size_list_real(map_int_real(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_5814_length__map,axiom,
    ! [F: fun_real_bool,Xs: list_real] : size_size_list_bool(map_real_bool(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5815_length__map,axiom,
    ! [F: fun_bool_bool,Xs: list_bool] : size_size_list_bool(map_bool_bool(F,Xs)) = size_size_list_bool(Xs) ).

% length_map
tff(fact_5816_length__map,axiom,
    ! [F: fun_nat_bool,Xs: list_nat] : size_size_list_bool(map_nat_bool(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5817_length__map,axiom,
    ! [F: fun_int_bool,Xs: list_int] : size_size_list_bool(map_int_bool(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_5818_length__map,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : size_size_list_nat(map_VEBT_VEBT_nat(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5819_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) )
    <=> ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F,X5) = aa_VEBT_VEBT_real(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_5820_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) )
    <=> ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F,X5) = aa_VEBT_VEBT_nat(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_5821_map__eq__conv,axiom,
    ! [F: fun_nat_bool,Xs: list_nat,G: fun_nat_bool] :
      ( ( map_nat_bool(F,Xs) = map_nat_bool(G,Xs) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => ( pp(aa_nat_bool(F,X5))
          <=> pp(aa_nat_bool(G,X5)) ) ) ) ).

% map_eq_conv
tff(fact_5822_map__eq__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_nat_nat] :
      ( ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) )
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Xs)))
         => ( aa_nat_nat(F,X5) = aa_nat_nat(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_5823_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F,Xs),N) = aa_VEB5832953974749077948T_VEBT(F,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_5824_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VE9120655849212850072T_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( nth_VEBT_VEBT(map_VE7998069337340375161T_VEBT(F,Xs),N) = aa_VEB4941942288398148609T_VEBT(F,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_5825_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VE712148701811721852_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),N) = aa_VEB3973023575446460973_VEBTi(F,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_5826_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VE8449866691444886503_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( nth_VEBT_VEBTi(map_VE483055756984248624_VEBTi(F,Xs),N) = aa_VEB5586453056816421800_VEBTi(F,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_5827_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBTi_nat(F,Xs)),N) = aa_VEBT_VEBTi_nat(F,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_5828_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_int(nth_int(map_VEBT_VEBT_int(F,Xs)),N) = aa_VEBT_VEBT_int(F,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_5829_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs)))
     => ( aa_nat_int(nth_int(map_VEBT_VEBTi_int(F,Xs)),N) = aa_VEBT_VEBTi_int(F,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_5830_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(F,Xs)),N) = aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_5831_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F,Xs)),N) = aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_5832_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: fun_real_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
     => ( nth_VEBT_VEBT(map_real_VEBT_VEBT(F,Xs),N) = aa_real_VEBT_VEBT(F,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_5833_list_Omap__ident,axiom,
    ! [T: list_nat] : map_nat_nat(aTP_Lamm_ar,T) = T ).

% list.map_ident
tff(fact_5834_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5835_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5836_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5837_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5838_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_bool_real,Ys: list_bool] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_bool_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5839_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_bool_nat,Ys: list_bool] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_bool_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_bool(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5840_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5841_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5842_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_int_real,Ys: list_int] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_int_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5843_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_int_nat,Ys: list_int] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_int_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5844_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( X = Ya )
     => ( ! [Z5: vEBT_VEBT] :
            ( member_VEBT_VEBT(Z5,set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_real(F,Z5) = aa_VEBT_VEBT_real(G,Z5) ) )
       => ( map_VEBT_VEBT_real(F,X) = map_VEBT_VEBT_real(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5845_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( X = Ya )
     => ( ! [Z5: vEBT_VEBT] :
            ( member_VEBT_VEBT(Z5,set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_nat(F,Z5) = aa_VEBT_VEBT_nat(G,Z5) ) )
       => ( map_VEBT_VEBT_nat(F,X) = map_VEBT_VEBT_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5846_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( ( X = Ya )
     => ( ! [Z5: nat] :
            ( pp(aa_set_nat_bool(member_nat(Z5),set_nat2(Ya)))
           => ( pp(aa_nat_bool(F,Z5))
            <=> pp(aa_nat_bool(G,Z5)) ) )
       => ( map_nat_bool(F,X) = map_nat_bool(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5847_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( X = Ya )
     => ( ! [Z5: nat] :
            ( pp(aa_set_nat_bool(member_nat(Z5),set_nat2(Ya)))
           => ( aa_nat_nat(F,Z5) = aa_nat_nat(G,Z5) ) )
       => ( map_nat_nat(F,X) = map_nat_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5848_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [Z5: vEBT_VEBT] :
          ( member_VEBT_VEBT(Z5,set_VEBT_VEBT2(X))
         => ( aa_VEBT_VEBT_real(F,Z5) = aa_VEBT_VEBT_real(G,Z5) ) )
     => ( map_VEBT_VEBT_real(F,X) = map_VEBT_VEBT_real(G,X) ) ) ).

% list.map_cong0
tff(fact_5849_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [Z5: vEBT_VEBT] :
          ( member_VEBT_VEBT(Z5,set_VEBT_VEBT2(X))
         => ( aa_VEBT_VEBT_nat(F,Z5) = aa_VEBT_VEBT_nat(G,Z5) ) )
     => ( map_VEBT_VEBT_nat(F,X) = map_VEBT_VEBT_nat(G,X) ) ) ).

% list.map_cong0
tff(fact_5850_list_Omap__cong0,axiom,
    ! [X: list_nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( ! [Z5: nat] :
          ( pp(aa_set_nat_bool(member_nat(Z5),set_nat2(X)))
         => ( pp(aa_nat_bool(F,Z5))
          <=> pp(aa_nat_bool(G,Z5)) ) )
     => ( map_nat_bool(F,X) = map_nat_bool(G,X) ) ) ).

% list.map_cong0
tff(fact_5851_list_Omap__cong0,axiom,
    ! [X: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [Z5: nat] :
          ( pp(aa_set_nat_bool(member_nat(Z5),set_nat2(X)))
         => ( aa_nat_nat(F,Z5) = aa_nat_nat(G,Z5) ) )
     => ( map_nat_nat(F,X) = map_nat_nat(G,X) ) ) ).

% list.map_cong0
tff(fact_5852_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa4: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Fa: fun_VEBT_VEBT_real] :
      ( ! [Z5: vEBT_VEBT,Za: vEBT_VEBT] :
          ( member_VEBT_VEBT(Z5,set_VEBT_VEBT2(X))
         => ( member_VEBT_VEBT(Za,set_VEBT_VEBT2(Xa4))
           => ( ( aa_VEBT_VEBT_real(F,Z5) = aa_VEBT_VEBT_real(Fa,Za) )
             => ( Z5 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_real(F,X) = map_VEBT_VEBT_real(Fa,Xa4) )
       => ( X = Xa4 ) ) ) ).

% list.inj_map_strong
tff(fact_5853_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa4: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Fa: fun_VEBT_VEBT_nat] :
      ( ! [Z5: vEBT_VEBT,Za: vEBT_VEBT] :
          ( member_VEBT_VEBT(Z5,set_VEBT_VEBT2(X))
         => ( member_VEBT_VEBT(Za,set_VEBT_VEBT2(Xa4))
           => ( ( aa_VEBT_VEBT_nat(F,Z5) = aa_VEBT_VEBT_nat(Fa,Za) )
             => ( Z5 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_nat(F,X) = map_VEBT_VEBT_nat(Fa,Xa4) )
       => ( X = Xa4 ) ) ) ).

% list.inj_map_strong
tff(fact_5854_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa4: list_nat,F: fun_nat_bool,Fa: fun_nat_bool] :
      ( ! [Z5: nat,Za: nat] :
          ( pp(aa_set_nat_bool(member_nat(Z5),set_nat2(X)))
         => ( pp(aa_set_nat_bool(member_nat(Za),set_nat2(Xa4)))
           => ( ( pp(aa_nat_bool(F,Z5))
              <=> pp(aa_nat_bool(Fa,Za)) )
             => ( Z5 = Za ) ) ) )
     => ( ( map_nat_bool(F,X) = map_nat_bool(Fa,Xa4) )
       => ( X = Xa4 ) ) ) ).

% list.inj_map_strong
tff(fact_5855_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa4: list_nat,F: fun_nat_nat,Fa: fun_nat_nat] :
      ( ! [Z5: nat,Za: nat] :
          ( pp(aa_set_nat_bool(member_nat(Z5),set_nat2(X)))
         => ( pp(aa_set_nat_bool(member_nat(Za),set_nat2(Xa4)))
           => ( ( aa_nat_nat(F,Z5) = aa_nat_nat(Fa,Za) )
             => ( Z5 = Za ) ) ) )
     => ( ( map_nat_nat(F,X) = map_nat_nat(Fa,Xa4) )
       => ( X = Xa4 ) ) ) ).

% list.inj_map_strong
tff(fact_5856_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F,X2) = aa_VEBT_VEBT_real(G,X2) ) )
     => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) ) ) ).

% map_ext
tff(fact_5857_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F,X2) = aa_VEBT_VEBT_nat(G,X2) ) )
     => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) ) ) ).

% map_ext
tff(fact_5858_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => ( pp(aa_nat_bool(F,X2))
          <=> pp(aa_nat_bool(G,X2)) ) )
     => ( map_nat_bool(F,Xs) = map_nat_bool(G,Xs) ) ) ).

% map_ext
tff(fact_5859_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => ( aa_nat_nat(F,X2) = aa_nat_nat(G,X2) ) )
     => ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) ) ) ).

% map_ext
tff(fact_5860_map__idI,axiom,
    ! [Xs: list_complex,F: fun_complex_complex] :
      ( ! [X2: complex] :
          ( member_complex(X2,set_complex2(Xs))
         => ( aa_complex_complex(F,X2) = X2 ) )
     => ( map_complex_complex(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5861_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F,X2) = X2 ) )
     => ( map_VE8901447254227204932T_VEBT(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5862_map__idI,axiom,
    ! [Xs: list_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => ( aa_real_real(F,X2) = X2 ) )
     => ( map_real_real(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5863_map__idI,axiom,
    ! [Xs: list_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => ( aa_nat_nat(F,X2) = X2 ) )
     => ( map_nat_nat(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5864_map__idI,axiom,
    ! [Xs: list_int,F: fun_int_int] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => ( aa_int_int(F,X2) = X2 ) )
     => ( map_int_int(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_5865_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xs = Ys )
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_real(F,X2) = aa_VEBT_VEBT_real(G,X2) ) )
       => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) ) ) ) ).

% map_cong
tff(fact_5866_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xs = Ys )
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_nat(F,X2) = aa_VEBT_VEBT_nat(G,X2) ) )
       => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_5867_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_bool,G: fun_nat_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Ys)))
           => ( pp(aa_nat_bool(F,X2))
            <=> pp(aa_nat_bool(G,X2)) ) )
       => ( map_nat_bool(F,Xs) = map_nat_bool(G,Ys) ) ) ) ).

% map_cong
tff(fact_5868_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xs = Ys )
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Ys)))
           => ( aa_nat_nat(F,X2) = aa_nat_nat(G,X2) ) )
       => ( map_nat_nat(F,Xs) = map_nat_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_5869_ex__map__conv,axiom,
    ! [Ys: list_bool,F: fun_nat_bool] :
      ( ? [Xs4: list_nat] : Ys = map_nat_bool(F,Xs4)
    <=> ! [X5: bool] :
          ( member_bool(X5,set_bool2(Ys))
         => ? [Xa3: nat] :
              ( pp(X5)
            <=> pp(aa_nat_bool(F,Xa3)) ) ) ) ).

% ex_map_conv
tff(fact_5870_ex__map__conv,axiom,
    ! [Ys: list_real,F: fun_VEBT_VEBT_real] :
      ( ? [Xs4: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_real(F,Xs4)
    <=> ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(Ys)))
         => ? [Xa3: vEBT_VEBT] : X5 = aa_VEBT_VEBT_real(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5871_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_VEBT_VEBT_nat] :
      ( ? [Xs4: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_nat(F,Xs4)
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Ys)))
         => ? [Xa3: vEBT_VEBT] : X5 = aa_VEBT_VEBT_nat(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5872_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_nat_nat] :
      ( ? [Xs4: list_nat] : Ys = map_nat_nat(F,Xs4)
    <=> ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(Ys)))
         => ? [Xa3: nat] : X5 = aa_nat_nat(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_5873_map__update,axiom,
    ! [F: fun_nat_bool,Xs: list_nat,K: nat,Y: nat] : map_nat_bool(F,list_update_nat(Xs,K,Y)) = list_update_bool(map_nat_bool(F,Xs),K,aa_nat_bool(F,Y)) ).

% map_update
tff(fact_5874_map__update,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,K: nat,Y: nat] : map_nat_nat(F,list_update_nat(Xs,K,Y)) = list_update_nat(map_nat_nat(F,Xs),K,aa_nat_nat(F,Y)) ).

% map_update
tff(fact_5875_map__update,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_update_real(map_VEBT_VEBT_real(F,Xs),K,aa_VEBT_VEBT_real(F,Y)) ).

% map_update
tff(fact_5876_map__update,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_update_nat(map_VEBT_VEBT_nat(F,Xs),K,aa_VEBT_VEBT_nat(F,Y)) ).

% map_update
tff(fact_5877_map__update,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_u1324408373059187874T_VEBT(map_VE8901447254227204932T_VEBT(F,Xs),K,aa_VEB5832953974749077948T_VEBT(F,Y)) ).

% map_update
tff(fact_5878_map__update,axiom,
    ! [F: fun_VE712148701811721852_VEBTi,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VE7029150624388687525_VEBTi(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_u6098035379799741383_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),K,aa_VEB3973023575446460973_VEBTi(F,Y)) ).

% map_update
tff(fact_5879_map__update,axiom,
    ! [F: fun_VE9120655849212850072T_VEBT,Xs: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] : map_VE7998069337340375161T_VEBT(F,list_u6098035379799741383_VEBTi(Xs,K,Y)) = list_u1324408373059187874T_VEBT(map_VE7998069337340375161T_VEBT(F,Xs),K,aa_VEB4941942288398148609T_VEBT(F,Y)) ).

% map_update
tff(fact_5880_map__update,axiom,
    ! [F: fun_VE8449866691444886503_VEBTi,Xs: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] : map_VE483055756984248624_VEBTi(F,list_u6098035379799741383_VEBTi(Xs,K,Y)) = list_u6098035379799741383_VEBTi(map_VE483055756984248624_VEBTi(F,Xs),K,aa_VEB5586453056816421800_VEBTi(F,Y)) ).

% map_update
tff(fact_5881_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,X: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_real(F,X) ) )
     => ( map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(L,I,X)) = map_VEBT_VEBT_real(F,L) ) ) ).

% map_upd_eq
tff(fact_5882_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,X: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)))
       => ( aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_nat(F,X) ) )
     => ( map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(L,I,X)) = map_VEBT_VEBT_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_5883_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_bool,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
       => ( pp(aa_nat_bool(F,aa_nat_nat(nth_nat(L),I)))
        <=> pp(aa_nat_bool(F,X)) ) )
     => ( map_nat_bool(F,list_update_nat(L,I,X)) = map_nat_bool(F,L) ) ) ).

% map_upd_eq
tff(fact_5884_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
       => ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(F,X) ) )
     => ( map_nat_nat(F,list_update_nat(L,I,X)) = map_nat_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_5885_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) ).

% VEBT_internal.cnt.simps(2)
tff(fact_5886_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,X) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_5887_vebt__buildupi__rule,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)),vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% vebt_buildupi_rule
tff(fact_5888_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),U))) ) ).

% htt_vebt_buildupi'_univ
tff(fact_5889_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) )
     => pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),U))) ) ).

% htt_vebt_buildupi_univ
tff(fact_5890_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_maxti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bq(T),Ti)),one_one_nat)) ).

% vebt_maxti_hT
tff(fact_5891_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] : pp(aa_nat_bool(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Ti),vEBT_vebt_minti(Ti),aa_VEB7611521244314475576t_assn(aTP_Lamm_bp(T),Ti)),one_one_nat)) ).

% vebt_minti_hT
tff(fact_5892_lowi__hT,axiom,
    ! [X: nat,N: nat] : pp(aa_nat_bool(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(X,N),aa_nat_fun_nat_assn(aTP_Lamm_bx(X),N)),one_one_nat)) ).

% lowi_hT
tff(fact_5893_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X2)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5894_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X2)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5895_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_bool,F: fun_bool_nat,Bound: nat,I: nat] :
      ( ! [X2: bool] :
          ( member_bool(X2,set_bool2(Xs))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(F,X2)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_bool_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5896_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,X2)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5897_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_int,F: fun_int_nat,Bound: nat,I: nat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X2)),Bound)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_int_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5898_real__nat__list,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_bz(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_5899_real__nat__list,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(aTP_Lamm_ca(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_5900_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% htt_vebt_buildupi'
tff(fact_5901_htt__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% htt_vebt_buildupi
tff(fact_5902_highi__hT,axiom,
    ! [X: nat,N: nat] : pp(aa_nat_bool(time_htt_nat(one_one_assn,vEBT_VEBT_highi(X,N),aa_nat_fun_nat_assn(aTP_Lamm_by(X),N)),one_one_nat)) ).

% highi_hT
tff(fact_5903_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_5904_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,X) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_5905_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_5906_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,X) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5907_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_5908_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,X) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5909_T__vebt__buildupi,axiom,
    ! [N: nat,H: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(N),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% T_vebt_buildupi
tff(fact_5910_lowi__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_lowi(X,N) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% lowi_def
tff(fact_5911_highi__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_highi(X,N) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% highi_def
tff(fact_5912_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ) ).

% TBOUND_buildupi
tff(fact_5913_lowsimp,axiom,
    ! [X: nat,N: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) = vEBT_VEBT_lowi(X,N) ).

% lowsimp
tff(fact_5914_highsimp,axiom,
    ! [X: nat,N: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)) = vEBT_VEBT_highi(X,N) ).

% highsimp
tff(fact_5915_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_V739175172307565963ildupi(N)),aa_nat_nat(vEBT_V441764108873111860ildupi,N))) ).

% TBOUND_vebt_buildupi
tff(fact_5916_return__sp__rule,axiom,
    ! [P: assn,X: bool] : hoare_2774773592644289642e_bool(P,heap_T5885414692442464636n_bool(X),aa_boo2938360491098410167l_assn(aTP_Lamm_cb(P),X)) ).

% return_sp_rule
tff(fact_5917_return__sp__rule,axiom,
    ! [P: assn,X: option_nat] : hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(X),aa_opt3861602675778688675t_assn(aTP_Lamm_cc(P),X)) ).

% return_sp_rule
tff(fact_5918_return__sp__rule,axiom,
    ! [P: assn,X: nat] : hoare_3067605981109127869le_nat(P,heap_Time_return_nat(X),aa_nat_fun_nat_assn(aTP_Lamm_cd(P),X)) ).

% return_sp_rule
tff(fact_5919_return__sp__rule,axiom,
    ! [P: assn,X: vEBT_VEBTi] : hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(X),aa_VEB1545114916007340237i_assn(aTP_Lamm_ce(P),X)) ).

% return_sp_rule
tff(fact_5920_time__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C: nat,N: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H3: heap_e7401611519738050253t_unit] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_time_VEBT_VEBTi(X,H3)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(N,X),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% time_replicate
tff(fact_5921_TBOUND__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_T5737551269749752165_VEBTi(X),C))
     => pp(aa_nat_bool(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(N,X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_5922_TBOUND__replicate,axiom,
    ! [X: heap_Time_Heap_nat,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_TBOUND_nat(X),C))
     => pp(aa_nat_bool(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(N,X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_5923_TBOUND__replicate,axiom,
    ! [X: heap_T2636463487746394924on_nat,C: nat,N: nat] :
      ( pp(aa_nat_bool(time_T8353473612707095248on_nat(X),C))
     => pp(aa_nat_bool(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(N,X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N)))) ) ).

% TBOUND_replicate
tff(fact_5924_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5925_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(aa_num_num(bit1,M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5926_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_5927_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(bit0(M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5055182867167087721od_nat(bit0(M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(bit0(M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5928_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(bit0(M),aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,bit0(M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique5052692396658037445od_int(bit0(M),aa_num_num(bit1,N)) = unique5024387138958732305ep_int(aa_num_num(bit1,N),unique5052692396658037445od_int(bit0(M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5929_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(bit0(M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(bit0(M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
       => ( unique3479559517661332726nteger(bit0(M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(bit0(M),bit0(aa_num_num(bit1,N)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5930_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5931_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5932_TBOUND__lowi,axiom,
    ! [X: nat,N: nat] : pp(aa_nat_bool(time_TBOUND_nat(vEBT_VEBT_lowi(X,N)),one_one_nat)) ).

% TBOUND_lowi
tff(fact_5933_TBOUND__highi,axiom,
    ! [X: nat,N: nat] : pp(aa_nat_bool(time_TBOUND_nat(vEBT_VEBT_highi(X,N)),one_one_nat)) ).

% TBOUND_highi
tff(fact_5934_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_vebt_minti(T)),one_one_nat)) ).

% TBOUND_vebt_minti
tff(fact_5935_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_vebt_maxti(T)),one_one_nat)) ).

% TBOUND_vebt_maxti
tff(fact_5936_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5937_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5938_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5939_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
    <=> pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,unique5052692396658037445od_int(N,M))) ) ).

% dvd_numeral_simp
tff(fact_5940_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
    <=> unique6322359934112328802ux_nat(unique5055182867167087721od_nat(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5941_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5942_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,bit0(N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5943_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,bit0(N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5944_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,bit0(N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5945_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5946_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5947_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5948_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_5949_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : vEBT_vebt_maxti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_maxti.simps(2)
tff(fact_5950_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : vEBT_vebt_minti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_minti.simps(2)
tff(fact_5951_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_5952_divmod__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod_def
tff(fact_5953_divmod__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% divmod_def
tff(fact_5954_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod'_nat_def
tff(fact_5955_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : vEBT_vebt_minti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz2)) = heap_T3487192422709364219on_nat(some_nat(Mi)) ).

% vebt_minti.simps(3)
tff(fact_5956_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : vEBT_vebt_maxti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz2)) = heap_T3487192422709364219on_nat(some_nat(Ma)) ).

% vebt_maxti.simps(3)
tff(fact_5957_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,bit0(N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5958_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique5052692396658037445od_int(M,N) = unique5024387138958732305ep_int(N,unique5052692396658037445od_int(M,bit0(N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5959_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
       => ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,bit0(N))) ) ) ) ).

% divmod_divmod_step
tff(fact_5960_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),numeral_numeral_real(bit0(one))) ).

% artanh_def
tff(fact_5961_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_T8145700208782473153_VEBTi,Q5: fun_VEBT_VEBTi_assn,T2: nat,P: assn,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(P2,C,Q5),T2))
     => ( entails(P,P2)
       => ( ! [X2: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q5,X2),aa_VEBT_VEBTi_assn(Q,X2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T2),T))
           => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_5962_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_T2636463487746394924on_nat,Q5: fun_option_nat_assn,T2: nat,P: assn,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(P2,C,Q5),T2))
     => ( entails(P,P2)
       => ( ! [X2: option_nat] : entails(aa_option_nat_assn(Q5,X2),aa_option_nat_assn(Q,X2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T2),T))
           => pp(aa_nat_bool(time_htt_option_nat(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_5963_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_Time_Heap_nat,Q5: fun_nat_assn,T2: nat,P: assn,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(P2,C,Q5),T2))
     => ( entails(P,P2)
       => ( ! [X2: nat] : entails(aa_nat_assn(Q5,X2),aa_nat_assn(Q,X2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T2),T))
           => pp(aa_nat_bool(time_htt_nat(P,C,Q),T)) ) ) ) ) ).

% htt_cons_rule
tff(fact_5964_nth__rule,axiom,
    ! [I: nat,Xs: list_int,A: array_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => hoare_3065115510600077593le_int(snga_assn_int(A,Xs),array_nth_int(A,I),aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_cf(I),Xs),A)) ) ).

% nth_rule
tff(fact_5965_nth__rule,axiom,
    ! [I: nat,Xs: list_bool,A: array_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => hoare_2774773592644289642e_bool(snga_assn_bool(A,Xs),array_nth_bool(A,I),aa_arr3483125744708554596l_assn(aa_lis2577335236791371354l_assn(aTP_Lamm_cg(I),Xs),A)) ) ).

% nth_rule
tff(fact_5966_nth__rule,axiom,
    ! [I: nat,Xs: list_option_nat,A: array_option_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6086282163384603972on_nat(Xs)))
     => hoare_7629718768684598413on_nat(snga_assn_option_nat(A,Xs),array_nth_option_nat(A,I),aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_ch(I),Xs),A)) ) ).

% nth_rule
tff(fact_5967_nth__rule,axiom,
    ! [I: nat,Xs: list_nat,A: array_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),array_nth_nat(A,I),aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ci(I),Xs),A)) ) ).

% nth_rule
tff(fact_5968_nth__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: array_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs)))
     => hoare_1429296392585015714_VEBTi(snga_assn_VEBT_VEBTi(A,Xs),array_nth_VEBT_VEBTi(A,I),aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_cj(I),Xs),A)) ) ).

% nth_rule
tff(fact_5969_return__cons__rule,axiom,
    ! [P: assn,Q: fun_bool_assn,X: bool] :
      ( entails(P,aa_bool_assn(Q,X))
     => hoare_2774773592644289642e_bool(P,heap_T5885414692442464636n_bool(X),Q) ) ).

% return_cons_rule
tff(fact_5970_return__cons__rule,axiom,
    ! [P: assn,Q: fun_option_nat_assn,X: option_nat] :
      ( entails(P,aa_option_nat_assn(Q,X))
     => hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(X),Q) ) ).

% return_cons_rule
tff(fact_5971_return__cons__rule,axiom,
    ! [P: assn,Q: fun_nat_assn,X: nat] :
      ( entails(P,aa_nat_assn(Q,X))
     => hoare_3067605981109127869le_nat(P,heap_Time_return_nat(X),Q) ) ).

% return_cons_rule
tff(fact_5972_return__cons__rule,axiom,
    ! [P: assn,Q: fun_VEBT_VEBTi_assn,X: vEBT_VEBTi] :
      ( entails(P,aa_VEBT_VEBTi_assn(Q,X))
     => hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(X),Q) ) ).

% return_cons_rule
tff(fact_5973_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,Ps: assn,F2: assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F2))
       => hoare_2774773592644289642e_bool(Ps,C,aa_ass2376637134808068204l_assn(aTP_Lamm_bt(Q),F2)) ) ) ).

% fi_rule
tff(fact_5974_fi__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ps: assn,F2: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F2))
       => hoare_7629718768684598413on_nat(Ps,C,aa_ass4112278883933122603t_assn(aTP_Lamm_bu(Q),F2)) ) ) ).

% fi_rule
tff(fact_5975_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,Ps: assn,F2: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F2))
       => hoare_3067605981109127869le_nat(Ps,C,aa_assn_fun_nat_assn(aTP_Lamm_bv(Q),F2)) ) ) ).

% fi_rule
tff(fact_5976_fi__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ps: assn,F2: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F2))
       => hoare_1429296392585015714_VEBTi(Ps,C,aa_ass4389913640333755968i_assn(aTP_Lamm_bw(Q),F2)) ) ) ).

% fi_rule
tff(fact_5977_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(plus_plus_int(A),one_one_int),B),product_Pair_int_int(Q2,R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),product_Pair_int_int(Q2,aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),R2)),one_one_int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5978_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5979_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5980_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5981_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_5982_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_5983_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_5984_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5985_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_5986_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_5987_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R2)))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q6,R4)))
       => ( Q2 = Q6 ) ) ) ).

% unique_quotient
tff(fact_5988_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R2)))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q6,R4)))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_5989_eucl__rel__int__by0,axiom,
    ! [K: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,zero_zero_int),product_Pair_int_int(zero_zero_int,K))) ).

% eucl_rel_int_by0
tff(fact_5990_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q2,R2)))
     => ( aa_int_int(modulo_modulo_int(K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_5991_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q2,R2)))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_5992_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_less_self
tff(fact_5993_log__def,axiom,
    ! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_5994_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_bound
tff(fact_5995_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_gt_zero
tff(fact_5996_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_5997_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5998_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_ge_zero
tff(fact_5999_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(times_times_int(Q2),L) )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q2,zero_zero_int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_6000_eucl__rel__int,axiom,
    ! [K: int,L: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L),aa_int_int(modulo_modulo_int(K),L)))) ).

% eucl_rel_int
tff(fact_6001_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),C)),B) ).

% assn_aci(10)
tff(fact_6002_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(A),aa_assn_assn(times_times_assn(B),C)) = aa_assn_assn(times_times_assn(B),aa_assn_assn(times_times_assn(A),C)) ).

% star_aci(3)
tff(fact_6003_star__aci_I2_J,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(times_times_assn(A),B) = aa_assn_assn(times_times_assn(B),A) ).

% star_aci(2)
tff(fact_6004_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C) = aa_assn_assn(times_times_assn(A),aa_assn_assn(times_times_assn(B),C)) ).

% star_assoc
tff(fact_6005_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,Q) ) ).

% is_entails
tff(fact_6006_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6007_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_6008_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_6009_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6010_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_6011_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(bit0(one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_6012_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_6013_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_6014_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_6015_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( ( B != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_6016_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q2,R2)))
    <=> ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(L),Q2)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_6017_fr__rot,axiom,
    ! [A2: assn,B2: assn,C5: assn] :
      ( entails(aa_assn_assn(times_times_assn(A2),B2),C5)
     => entails(aa_assn_assn(times_times_assn(B2),A2),C5) ) ).

% fr_rot
tff(fact_6018_fr__refl,axiom,
    ! [A2: assn,B2: assn,C5: assn] :
      ( entails(A2,B2)
     => entails(aa_assn_assn(times_times_assn(A2),C5),aa_assn_assn(times_times_assn(B2),C5)) ) ).

% fr_refl
tff(fact_6019_fr__rot__rhs,axiom,
    ! [A2: assn,B2: assn,C5: assn] :
      ( entails(A2,aa_assn_assn(times_times_assn(B2),C5))
     => entails(A2,aa_assn_assn(times_times_assn(C5),B2)) ) ).

% fr_rot_rhs
tff(fact_6020_ent__frame__fwd,axiom,
    ! [P: assn,R3: assn,Ps: assn,F2: assn,Q: assn] :
      ( entails(P,R3)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F2))
       => ( entails(aa_assn_assn(times_times_assn(R3),F2),Q)
         => entails(Ps,Q) ) ) ) ).

% ent_frame_fwd
tff(fact_6021_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(A),one_one_assn) = A ).

% norm_assertion_simps(2)
tff(fact_6022_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(one_one_assn),A) = A ).

% norm_assertion_simps(1)
tff(fact_6023_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6024_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_bool,Q: fun_bool_assn,R3: assn] :
      ( hoare_2774773592644289642e_bool(P,C,Q)
     => hoare_2774773592644289642e_bool(aa_assn_assn(times_times_assn(R3),P),C,aa_ass2376637134808068204l_assn(aTP_Lamm_ck(Q),R3)) ) ).

% frame_rule_left
tff(fact_6025_frame__rule__left,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R3: assn] :
      ( hoare_7629718768684598413on_nat(P,C,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(R3),P),C,aa_ass4112278883933122603t_assn(aTP_Lamm_cl(Q),R3)) ) ).

% frame_rule_left
tff(fact_6026_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: fun_nat_assn,R3: assn] :
      ( hoare_3067605981109127869le_nat(P,C,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(R3),P),C,aa_assn_fun_nat_assn(aTP_Lamm_cm(Q),R3)) ) ).

% frame_rule_left
tff(fact_6027_frame__rule__left,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R3: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(R3),P),C,aa_ass4389913640333755968i_assn(aTP_Lamm_cn(Q),R3)) ) ).

% frame_rule_left
tff(fact_6028_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R2)))
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),product_Pair_int_int(Q2,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),R2))))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_6029_mod__frame__fwd,axiom,
    ! [Ps: assn,H: produc3658429121746597890et_nat,P: assn,R3: assn,F2: assn] :
      ( rep_assn(Ps,H)
     => ( entails(P,R3)
       => ( entails(Ps,aa_assn_assn(times_times_assn(P),F2))
         => rep_assn(aa_assn_assn(times_times_assn(R3),F2),H) ) ) ) ).

% mod_frame_fwd
tff(fact_6030_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_6031_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_option_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_6032_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: bool,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(aa_assn_assn(times_times_assn(P),aa_bool_assn(pure_assn,B)),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_6033_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_VEBT_VEBTi(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_VEBT_VEBTi(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_6034_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_option_nat(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_option_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_6035_norm__pre__pure__iff__htt_H,axiom,
    ! [B: bool,P: assn,F: heap_Time_Heap_nat,Q: fun_nat_assn,T: nat] :
      ( pp(aa_nat_bool(time_htt_nat(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,B)),P),F,Q),T))
    <=> ( pp(B)
       => pp(aa_nat_bool(time_htt_nat(P,F,Q),T)) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_6036_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(product_Pair_int_int(Q2,R2)) = aa_int_int(plus_plus_int(Q2),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_6037_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6038_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys))))
     => ( nth_Pr316670251186196177_VEBTi(produc316462671093861988_VEBTi(Xs,Ys),N) = produc6084888613844515218_VEBTi(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6039_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_s6755466524823107622T_VEBT(Ys))))
     => ( nth_Pr8725177398587324397T_VEBT(produc1285381384045549624T_VEBT(Xs,Ys),N) = produc7053807326796202854T_VEBT(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6040_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_s7982070591426661849_VEBTi(Ys))))
     => ( nth_Pr6329974346453275474_VEBTi(produc194614972289024177_VEBTi(Xs,Ys),N) = produc436343169921013763_VEBTi(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6041_product__nth,axiom,
    ! [N: nat,Xs: list_num,Ys: list_num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys))))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),N) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_6042_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys))))
     => ( nth_Pr6842391030413306568T_real(produc4908677263432625371T_real(Xs,Ys),N) = produc8117437818029410057T_real(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6043_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_real(Ys))))
     => ( nth_Pr3433448822664029129i_real(produc5476717833281694120i_real(Xs,Ys),N) = produc8457151488442208762i_real(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6044_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys),N) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_6045_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_bool(Ys))))
     => ( nth_Pr1878200693239305296i_bool(produc9101113540930457115i_bool(Xs,Ys),N) = produc7282046070707253577i_bool(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).

% product_nth
tff(fact_6046_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_6047_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6048_vebt__maxti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leafi(A4,B4) )
           => ~ ( ( pp(B4)
                 => ( Y = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                & ( ~ pp(B4)
                 => ( ( pp(A4)
                     => ( Y = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                    & ( ~ pp(A4)
                     => ( Y = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz)
               => ( Y != heap_T3487192422709364219on_nat(some_nat(Ma2)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_6049_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_6050_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_6051_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_6052_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = uminus1351360451143612070nteger(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_6053_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_6054_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_6055_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_6056_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_6057_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(A)) = A ).

% add.inverse_inverse
tff(fact_6058_add_Oinverse__inverse,axiom,
    ! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).

% add.inverse_inverse
tff(fact_6059_VEBTi_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leafi(X21,X222) = vEBT_Leafi(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBTi.inject(2)
tff(fact_6060_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_6061_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_6062_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_6063_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_6064_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_6065_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_6066_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = uminus1351360451143612070nteger(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_6067_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = uminus_uminus_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_6068_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_6069_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_6070_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_6071_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_6072_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_6073_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_6074_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_6075_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_6076_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = uminus1351360451143612070nteger(A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_6077_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_6078_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_6079_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_6080_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_6081_add_Oinverse__neutral,axiom,
    uminus1351360451143612070nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_6082_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_6083_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_6084_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% neg_le_iff_le
tff(fact_6085_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% neg_le_iff_le
tff(fact_6086_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% neg_le_iff_le
tff(fact_6087_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% neg_less_iff_less
tff(fact_6088_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% neg_less_iff_less
tff(fact_6089_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_6090_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_6091_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6092_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6093_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6094_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(M)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6095_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(M)) = uminus_uminus_rat(numeral_numeral_rat(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6096_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_6097_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_6098_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_6099_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B)) = B ).

% add_minus_cancel
tff(fact_6100_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(uminus_uminus_rat(A),B)) = B ).

% add_minus_cancel
tff(fact_6101_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(plus_plus_int(A),B)) = B ).

% minus_add_cancel
tff(fact_6102_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_6103_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_6104_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_6105_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),plus_plus_rat(A,B)) = B ).

% minus_add_cancel
tff(fact_6106_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_6107_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_6108_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_6109_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) ).

% minus_add_distrib
tff(fact_6110_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
tff(fact_6111_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_right
tff(fact_6112_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_6113_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_6114_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ).

% mult_minus_right
tff(fact_6115_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_6116_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(times_times_int(A),B) ).

% minus_mult_minus
tff(fact_6117_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_6118_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_6119_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(A)),uminus1351360451143612070nteger(B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) ).

% minus_mult_minus
tff(fact_6120_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_6121_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_left
tff(fact_6122_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_6123_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_6124_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(A)),B) = uminus1351360451143612070nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ).

% mult_minus_left
tff(fact_6125_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_6126_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(minus_minus_int(B),A) ).

% minus_diff_eq
tff(fact_6127_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_6128_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_6129_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_6130_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).

% minus_diff_eq
tff(fact_6131_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% div_minus_minus
tff(fact_6132_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_6133_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6134_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6135_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(uminus1482373934393186551omplex,X)),Y))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6136_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,uminus1351360451143612070nteger(X)),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6137_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,uminus_uminus_rat(X)),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% minus_dvd_iff
tff(fact_6138_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6139_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6140_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_complex_complex(uminus1482373934393186551omplex,Y)))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6141_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),uminus1351360451143612070nteger(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6142_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),uminus_uminus_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% dvd_minus_iff
tff(fact_6143_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_6144_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_6145_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6146_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6147_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6148_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_6149_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_6150_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_6151_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_6152_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_6153_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_6154_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_le_0_iff_le
tff(fact_6155_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_6156_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_6157_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_6158_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_6159_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_6160_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_6161_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_6162_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_6163_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_6164_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_0_iff_less
tff(fact_6165_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_6166_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_6167_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_6168_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_6169_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_6170_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_6171_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_6172_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_pos
tff(fact_6173_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_6174_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_6175_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_6176_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_6177_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_6178_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_6179_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_6180_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_6181_ab__left__minus,axiom,
    ! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_6182_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_6183_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_6184_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_6185_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_6186_add_Oright__inverse,axiom,
    ! [A: rat] : plus_plus_rat(A,uminus_uminus_rat(A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_6187_diff__0,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_6188_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_6189_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_6190_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = uminus1351360451143612070nteger(A) ).

% diff_0
tff(fact_6191_diff__0,axiom,
    ! [A: rat] : minus_minus_rat(zero_zero_rat,A) = uminus_uminus_rat(A) ).

% diff_0
tff(fact_6192_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(minus_minus_int(zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_6193_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_6194_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_6195_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = uminus1351360451143612070nteger(B) ).

% verit_minus_simplify(3)
tff(fact_6196_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : minus_minus_rat(zero_zero_rat,B) = uminus_uminus_rat(B) ).

% verit_minus_simplify(3)
tff(fact_6197_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_6198_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6199_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6200_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = uminus1351360451143612070nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6201_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6202_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_6203_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_6204_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_6205_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Z) = uminus1351360451143612070nteger(Z) ).

% mult_minus1
tff(fact_6206_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_6207_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_6208_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_6209_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_6210_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(Z) ).

% mult_minus1_right
tff(fact_6211_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_6212_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(plus_plus_int(A),B) ).

% diff_minus_eq_add
tff(fact_6213_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_6214_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_6215_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,uminus1351360451143612070nteger(B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_6216_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
tff(fact_6217_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(B),A) ).

% uminus_add_conv_diff
tff(fact_6218_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_6219_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_6220_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_6221_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
tff(fact_6222_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_6223_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(A) ).

% div_minus1_right
tff(fact_6224_divide__minus1,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_6225_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_6226_divide__minus1,axiom,
    ! [X: rat] : divide_divide_rat(X,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(X) ).

% divide_minus1
tff(fact_6227_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_6228_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) ).

% minus_mod_self1
tff(fact_6229_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(N,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_6230_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_6231_of__int__minus,axiom,
    ! [Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_minus
tff(fact_6232_of__int__minus,axiom,
    ! [Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,Z)) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_minus
tff(fact_6233_of__int__minus,axiom,
    ! [Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,Z)) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_minus
tff(fact_6234_of__int__minus,axiom,
    ! [Z: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z)) = uminus1351360451143612070nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ).

% of_int_minus
tff(fact_6235_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_6236_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_6237_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_6238_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_6239_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K))) ).

% dbl_simps(1)
tff(fact_6240_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_6241_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K))) = uminus1351360451143612070nteger(neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_6242_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)
tff(fact_6243_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] : size_s3932428310213730859l_real(product_real_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6244_length__product,axiom,
    ! [Xs: list_real,Ys: list_bool] : size_s2850477280887376644l_bool(product_real_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_6245_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] : size_s1877336372972134351al_nat(product_real_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6246_length__product,axiom,
    ! [Xs: list_real,Ys: list_int] : size_s8610625264895183403al_int(product_real_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_6247_length__product,axiom,
    ! [Xs: list_bool,Ys: list_real] : size_s8805918724536545814l_real(product_bool_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6248_length__product,axiom,
    ! [Xs: list_bool,Ys: list_bool] : size_s6988337448923680857l_bool(product_bool_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_6249_length__product,axiom,
    ! [Xs: list_bool,Ys: list_nat] : size_s3243772863075988794ol_nat(product_bool_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6250_length__product,axiom,
    ! [Xs: list_bool,Ys: list_int] : size_s753689718144262038ol_int(product_bool_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_6251_length__product,axiom,
    ! [Xs: list_nat,Ys: list_real] : size_s7910714270633306959t_real(product_nat_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6252_length__product,axiom,
    ! [Xs: list_nat,Ys: list_bool] : size_s783543119881238752t_bool(product_nat_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_bool(Ys)) ).

% length_product
tff(fact_6253_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(plus_plus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_6254_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_6255_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_6256_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)
tff(fact_6257_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)
tff(fact_6258_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_6259_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_6260_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_6261_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)
tff(fact_6262_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)
tff(fact_6263_diff__numeral__special_I12_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_6264_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_6265_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_6266_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_6267_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)
tff(fact_6268_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6269_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6270_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6271_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6272_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(N)) = uminus_uminus_rat(one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6273_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6274_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6275_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6276_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( uminus1351360451143612070nteger(one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6277_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6278_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6279_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6280_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6281_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6282_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6283_minus__one__mult__self,axiom,
    ! [N: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_6284_minus__one__mult__self,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).

% minus_one_mult_self
tff(fact_6285_minus__one__mult__self,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).

% minus_one_mult_self
tff(fact_6286_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_6287_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_6288_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_6289_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_6290_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_6291_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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)
tff(fact_6292_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = numeral_numeral_real(U) ) ) ) ).

% max_number_of(2)
tff(fact_6293_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(2)
tff(fact_6294_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_6295_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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)
tff(fact_6296_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).

% max_number_of(3)
tff(fact_6297_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(3)
tff(fact_6298_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(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)
tff(fact_6299_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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)
tff(fact_6300_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).

% max_number_of(4)
tff(fact_6301_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(4)
tff(fact_6302_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_6303_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_6304_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6305_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6306_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6307_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W)),Y)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_6308_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(W)),Y)) = plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_6309_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6310_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6311_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6312_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6313_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6314_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6315_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6316_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6317_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6318_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6319_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_6320_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6321_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6322_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6323_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6324_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6325_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6326_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6327_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6328_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6329_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6330_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6331_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6332_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6333_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6334_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6335_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6336_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6337_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6338_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6339_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6340_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6341_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6342_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6343_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6344_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6345_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6346_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6347_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6348_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6349_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6350_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_6351_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_6352_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_6353_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).

% neg_numeral_le_iff
tff(fact_6354_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_6355_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_6356_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_6357_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).

% neg_numeral_less_iff
tff(fact_6358_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6359_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_6360_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6361_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6362_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6363_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6364_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6365_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6366_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6367_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6368_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> ( ( ( uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) = B ) )
        & ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6369_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ) )
        & ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6370_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6371_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) = A )
    <=> ( ( ( uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) ) )
        & ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6372_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6373_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6374_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6375_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6376_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6377_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6378_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6379_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6380_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_minus
tff(fact_6381_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_minus
tff(fact_6382_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_minus
tff(fact_6383_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_minus
tff(fact_6384_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_minus
tff(fact_6385_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_6386_ceiling__neg__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_6387_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_6388_round__neg__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_6389_round__neg__numeral,axiom,
    ! [N: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_6390_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_6391_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_6392_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_6393_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)
tff(fact_6394_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)
tff(fact_6395_diff__numeral__special_I10_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(one))) ).

% diff_numeral_special(10)
tff(fact_6396_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_6397_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_6398_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)
tff(fact_6399_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)
tff(fact_6400_diff__numeral__special_I11_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% diff_numeral_special(11)
tff(fact_6401_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_6402_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_6403_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = numera6620942414471956472nteger(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_6404_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)
tff(fact_6405_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_6406_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
tff(fact_6407_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_6408_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
tff(fact_6409_minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_6410_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_6411_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6412_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6413_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6414_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6415_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6416_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6417_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6418_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6419_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6420_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6421_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ).

% power_minus_odd
tff(fact_6422_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ).

% power_minus_odd
tff(fact_6423_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ).

% power_minus_odd
tff(fact_6424_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).

% power_minus_odd
tff(fact_6425_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),N)) ) ) ).

% power_minus_odd
tff(fact_6426_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6427_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6428_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6429_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6430_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6431_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6432_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6433_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6434_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6435_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6436_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),zero_zero_int))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),uminus_uminus_rat(one_one_rat))) ) ).

% ceiling_less_zero
tff(fact_6437_ceiling__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).

% ceiling_less_zero
tff(fact_6438_zero__le__ceiling,axiom,
    ! [X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).

% zero_le_ceiling
tff(fact_6439_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),X)) ) ).

% zero_le_ceiling
tff(fact_6440_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N))) ).

% numeral_div_minus_numeral
tff(fact_6441_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N))) ).

% minus_numeral_div_numeral
tff(fact_6442_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(one))) ).

% dbl_simps(4)
tff(fact_6443_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% dbl_simps(4)
tff(fact_6444_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(bit0(one))) ).

% dbl_simps(4)
tff(fact_6445_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(bit0(one))) ).

% dbl_simps(4)
tff(fact_6446_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)
tff(fact_6447_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_6448_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = one_one_int ).

% power_minus1_even
tff(fact_6449_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = one_one_real ).

% power_minus1_even
tff(fact_6450_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = one_one_complex ).

% power_minus1_even
tff(fact_6451_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_6452_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_6453_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_6454_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_6455_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_6456_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_6457_neg__one__even__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_6458_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_6459_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_6460_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_6461_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N) = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_6462_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_6463_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% signed_take_bit_0
tff(fact_6464_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% signed_take_bit_0
tff(fact_6465_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6466_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6467_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6468_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(X))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6469_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6470_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6471_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6472_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6473_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6474_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6475_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_6476_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),uminus_uminus_rat(numeral_numeral_rat(V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_6477_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)))) ) ).

% ceiling_le_neg_numeral
tff(fact_6478_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_6479_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(V))),X)) ) ).

% neg_numeral_less_ceiling
tff(fact_6480_minus__one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_6481_one__div__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_6482_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat))) ) ).

% ceiling_less_neg_numeral
tff(fact_6483_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real))) ) ).

% ceiling_less_neg_numeral
tff(fact_6484_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_6485_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(X)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),X)) ) ).

% neg_numeral_le_ceiling
tff(fact_6486_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6487_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6488_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6489_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6490_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6491_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6492_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6493_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6494_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6495_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6496_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6497_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6498_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6499_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6500_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6501_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6502_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_6503_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_6504_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_6505_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( uminus1351360451143612070nteger(B) = A ) ) ).

% minus_equation_iff
tff(fact_6506_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( uminus_uminus_rat(B) = A ) ) ).

% minus_equation_iff
tff(fact_6507_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_6508_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_6509_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_6510_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% equation_minus_iff
tff(fact_6511_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% equation_minus_iff
tff(fact_6512_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))) ) ).

% le_imp_neg_le
tff(fact_6513_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))) ) ).

% le_imp_neg_le
tff(fact_6514_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_6515_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_6516_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),A)) ) ).

% minus_le_iff
tff(fact_6517_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),A)) ) ).

% minus_le_iff
tff(fact_6518_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_le_iff
tff(fact_6519_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_le_iff
tff(fact_6520_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),uminus1351360451143612070nteger(A))) ) ).

% le_minus_iff
tff(fact_6521_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),uminus_uminus_rat(A))) ) ).

% le_minus_iff
tff(fact_6522_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_6523_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_6524_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6525_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6526_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6527_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))) ) ).

% verit_negate_coefficient(2)
tff(fact_6528_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_less_iff
tff(fact_6529_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_less_iff
tff(fact_6530_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),A)) ) ).

% minus_less_iff
tff(fact_6531_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),A)) ) ).

% minus_less_iff
tff(fact_6532_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_6533_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_6534_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),uminus1351360451143612070nteger(A))) ) ).

% less_minus_iff
tff(fact_6535_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),uminus_uminus_rat(A))) ) ).

% less_minus_iff
tff(fact_6536_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% numeral_neq_neg_numeral
tff(fact_6537_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numeral_numeral_real(M) != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% numeral_neq_neg_numeral
tff(fact_6538_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% numeral_neq_neg_numeral
tff(fact_6539_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6620942414471956472nteger(M) != uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) ).

% numeral_neq_neg_numeral
tff(fact_6540_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numeral_numeral_rat(M) != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% numeral_neq_neg_numeral
tff(fact_6541_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).

% neg_numeral_neq_numeral
tff(fact_6542_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) != numeral_numeral_real(N) ).

% neg_numeral_neq_numeral
tff(fact_6543_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N) ).

% neg_numeral_neq_numeral
tff(fact_6544_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : uminus1351360451143612070nteger(numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(N) ).

% neg_numeral_neq_numeral
tff(fact_6545_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : uminus_uminus_rat(numeral_numeral_rat(M)) != numeral_numeral_rat(N) ).

% neg_numeral_neq_numeral
tff(fact_6546_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_6547_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_6548_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_6549_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( uminus1351360451143612070nteger(A2) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),uminus1351360451143612070nteger(A)) ) ) ).

% group_cancel.neg1
tff(fact_6550_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( uminus_uminus_rat(A2) = plus_plus_rat(uminus_uminus_rat(K),uminus_uminus_rat(A)) ) ) ).

% group_cancel.neg1
tff(fact_6551_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_6552_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_6553_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_6554_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
tff(fact_6555_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
tff(fact_6556_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_6557_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_6558_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_6559_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)
tff(fact_6560_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)
tff(fact_6561_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_6562_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_6563_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_6564_one__neq__neg__one,axiom,
    one_one_Code_integer != uminus1351360451143612070nteger(one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_6565_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_6566_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_6567_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_6568_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_6569_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(A)),B) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),uminus1351360451143612070nteger(B)) ).

% minus_mult_commute
tff(fact_6570_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) ).

% minus_mult_commute
tff(fact_6571_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),A) = aa_int_int(times_times_int(B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_6572_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_6573_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_6574_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),B) )
    <=> ( ( A = B )
        | ( A = uminus1351360451143612070nteger(B) ) ) ) ).

% square_eq_iff
tff(fact_6575_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = uminus_uminus_rat(B) ) ) ) ).

% square_eq_iff
tff(fact_6576_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_6577_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_6578_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_6579_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(B),A) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),B) ).

% minus_diff_commute
tff(fact_6580_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
tff(fact_6581_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) ).

% minus_diff_minus
tff(fact_6582_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% minus_diff_minus
tff(fact_6583_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_6584_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_6585_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
tff(fact_6586_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_6587_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,uminus1351360451143612070nteger(B)) = divide6298287555418463151nteger(uminus1351360451143612070nteger(A),B) ).

% div_minus_right
tff(fact_6588_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_6589_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_6590_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
tff(fact_6591_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_6592_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_6593_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
tff(fact_6594_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_6595_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_6596_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
tff(fact_6597_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_6598_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B)) ).

% mod_minus_right
tff(fact_6599_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A3),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_6600_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
     => ( modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A3),B) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_6601_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_6602_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) ).

% mod_minus_eq
tff(fact_6603_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_6604_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% int_cases2
tff(fact_6605_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(N),K))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_6606_VEBTi_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,vEBT_Leafi(X21,X222)) = zero_zero_nat ).

% VEBTi.size(4)
tff(fact_6607_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_6608_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_6609_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_6610_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_6611_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus_uminus_rat(numeral_numeral_rat(K)) ).

% of_int_neg_numeral
tff(fact_6612_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] : Y != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X112,X122,X132),X142)
     => ~ ! [X212: bool,X223: bool] : Y != vEBT_Leafi(X212,X223) ) ).

% VEBTi.exhaust
tff(fact_6613_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: bool,X222: bool] : aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14) != vEBT_Leafi(X21,X222) ).

% VEBTi.distinct(1)
tff(fact_6614_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_6615_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% zero_neq_neg_numeral
tff(fact_6616_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_6617_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_z3403309356797280102nteger != uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) ).

% zero_neq_neg_numeral
tff(fact_6618_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% zero_neq_neg_numeral
tff(fact_6619_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_6620_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_6621_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_numeral_le_neg_numeral
tff(fact_6622_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_le_neg_numeral
tff(fact_6623_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_le_numeral
tff(fact_6624_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N))) ).

% neg_numeral_le_numeral
tff(fact_6625_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N))) ).

% neg_numeral_le_numeral
tff(fact_6626_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_le_numeral
tff(fact_6627_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_numeral_less_neg_numeral
tff(fact_6628_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_6629_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_6630_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_numeral_less_neg_numeral
tff(fact_6631_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).

% neg_numeral_less_numeral
tff(fact_6632_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N))) ).

% neg_numeral_less_numeral
tff(fact_6633_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).

% neg_numeral_less_numeral
tff(fact_6634_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N))) ).

% neg_numeral_less_numeral
tff(fact_6635_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6636_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6637_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6638_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
tff(fact_6639_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
tff(fact_6640_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6641_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6642_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6643_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
tff(fact_6644_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
tff(fact_6645_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_6646_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_6647_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_6648_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( uminus1351360451143612070nteger(A) = B ) ) ).

% add.inverse_unique
tff(fact_6649_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = zero_zero_rat )
     => ( uminus_uminus_rat(A) = B ) ) ).

% add.inverse_unique
tff(fact_6650_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_6651_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_6652_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_6653_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
tff(fact_6654_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
tff(fact_6655_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_6656_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_6657_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_6658_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% add_eq_0_iff
tff(fact_6659_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
tff(fact_6660_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_6661_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_6662_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_6663_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != uminus1351360451143612070nteger(one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_6664_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_6665_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_6666_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_6667_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_6668_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_6669_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_6670_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_6671_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_6672_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_6673_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_6674_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_6675_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_6676_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_6677_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_6678_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_6679_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_6680_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_6681_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_6682_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% one_neq_neg_numeral
tff(fact_6683_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_6684_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != uminus1351360451143612070nteger(numera6620942414471956472nteger(N)) ).

% one_neq_neg_numeral
tff(fact_6685_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% one_neq_neg_numeral
tff(fact_6686_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_6687_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_real(N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_6688_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_6689_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6620942414471956472nteger(N) != uminus1351360451143612070nteger(one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_6690_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_rat(N) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_6691_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(times_times_int(X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_6692_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% numeral_times_minus_swap
tff(fact_6693_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(X),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_6694_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(W)),uminus1351360451143612070nteger(X)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),uminus1351360451143612070nteger(numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_6695_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),uminus_uminus_rat(X)) = aa_rat_rat(times_times_rat(X),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_6696_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6697_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6698_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
tff(fact_6699_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6700_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6701_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
tff(fact_6702_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(times_times_int(X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_6703_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_6704_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_6705_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_6706_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_6707_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6708_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6709_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6710_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6711_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6712_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_6713_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_6714_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_6715_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% diff_conv_add_uminus
tff(fact_6716_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).

% diff_conv_add_uminus
tff(fact_6717_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(minus_minus_int(A),B2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6718_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6719_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6720_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_6721_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = plus_plus_rat(K,B) )
     => ( minus_minus_rat(A,B2) = plus_plus_rat(uminus_uminus_rat(K),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_6722_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_6723_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_6724_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_6725_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(uminus1351360451143612070nteger(A),B) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_6726_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( divide_divide_rat(uminus_uminus_rat(A),B) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_neg_div
tff(fact_6727_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_6728_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_6729_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_6730_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_6731_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( divide_divide_rat(A,uminus_uminus_rat(B)) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_div_neg
tff(fact_6732_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_6733_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z: int] :
      ( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
       => pp(aa_int_bool(P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_6734_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(times_times_int(M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_6735_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(times_times_int(M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_6736_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(minus_minus_int(zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_6737_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_6738_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_6739_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% not_int_zless_negative
tff(fact_6740_vebt__buildupi_Osimps_I1_J,axiom,
    vEBT_vebt_buildupi(zero_zero_nat) = heap_T3630416162098727440_VEBTi(vEBT_Leafi(fFalse,fFalse)) ).

% vebt_buildupi.simps(1)
tff(fact_6741_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V739175172307565963ildupi(zero_zero_nat) = heap_T3630416162098727440_VEBTi(vEBT_Leafi(fFalse,fFalse)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_6742_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_zero_le_neg_numeral
tff(fact_6743_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_zero_le_neg_numeral
tff(fact_6744_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_zero_le_neg_numeral
tff(fact_6745_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_le_neg_numeral
tff(fact_6746_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_6747_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_6748_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_6749_neg__numeral__le__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_6750_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).

% not_zero_less_neg_numeral
tff(fact_6751_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).

% not_zero_less_neg_numeral
tff(fact_6752_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(N)))) ).

% not_zero_less_neg_numeral
tff(fact_6753_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N)))) ).

% not_zero_less_neg_numeral
tff(fact_6754_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_6755_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_6756_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_6757_neg__numeral__less__zero,axiom,
    ! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_6758_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_6759_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_6760_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_6761_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_6762_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_6763_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_6764_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_6765_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_6766_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_6767_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_6768_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_6769_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_6770_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_6771_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_6772_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_6773_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_6774_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M)))) ).

% not_one_le_neg_numeral
tff(fact_6775_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M)))) ).

% not_one_le_neg_numeral
tff(fact_6776_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).

% not_one_le_neg_numeral
tff(fact_6777_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_le_neg_numeral
tff(fact_6778_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_6779_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_6780_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_6781_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_6782_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_6783_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_6784_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_6785_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_6786_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_le_numeral
tff(fact_6787_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M))) ).

% neg_one_le_numeral
tff(fact_6788_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M))) ).

% neg_one_le_numeral
tff(fact_6789_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_le_numeral
tff(fact_6790_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_6791_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_6792_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_6793_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_6794_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_6795_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_6796_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_6797_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_6798_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_less_neg_numeral
tff(fact_6799_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).

% not_one_less_neg_numeral
tff(fact_6800_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M)))) ).

% not_one_less_neg_numeral
tff(fact_6801_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M)))) ).

% not_one_less_neg_numeral
tff(fact_6802_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_6803_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_6804_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_6805_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_6806_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_less_numeral
tff(fact_6807_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M))) ).

% neg_one_less_numeral
tff(fact_6808_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_less_numeral
tff(fact_6809_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M))) ).

% neg_one_less_numeral
tff(fact_6810_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_6811_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_6812_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_6813_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_6814_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_6815_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_6816_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6817_uminus__numeral__One,axiom,
    uminus1351360451143612070nteger(numera6620942414471956472nteger(one)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6818_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_6819_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_6820_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_6821_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_6822_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(one))),B) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(1)
tff(fact_6823_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(1)
tff(fact_6824_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(times_times_int(B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_6825_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_6826_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_6827_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),uminus1351360451143612070nteger(numera6620942414471956472nteger(one))) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(2)
tff(fact_6828_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(2)
tff(fact_6829_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6830_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6831_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
tff(fact_6832_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6833_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6834_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)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = uminus_uminus_rat(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6835_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6836_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6837_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) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6838_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_6839_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_6840_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) = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_6841_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_6842_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_6843_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = uminus_uminus_rat(divide_divide_rat(B,C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = uminus_uminus_rat(B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_6844_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).

% power_minus
tff(fact_6845_power__minus,axiom,
    ! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% power_minus
tff(fact_6846_power__minus,axiom,
    ! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(A),N)) ).

% power_minus
tff(fact_6847_power__minus,axiom,
    ! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% power_minus
tff(fact_6848_power__minus,axiom,
    ! [A: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).

% power_minus
tff(fact_6849_power__minus__Bit0,axiom,
    ! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(K))) ).

% power_minus_Bit0
tff(fact_6850_power__minus__Bit0,axiom,
    ! [X: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(K))) ).

% power_minus_Bit0
tff(fact_6851_power__minus__Bit0,axiom,
    ! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(K))) ).

% power_minus_Bit0
tff(fact_6852_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(X)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(K))) ).

% power_minus_Bit0
tff(fact_6853_power__minus__Bit0,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,bit0(K))) = aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(K))) ).

% power_minus_Bit0
tff(fact_6854_power__minus__Bit1,axiom,
    ! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6855_power__minus__Bit1,axiom,
    ! [X: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6856_power__minus__Bit1,axiom,
    ! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6857_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6858_power__minus__Bit1,axiom,
    ! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6859_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_6860_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_6861_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_6862_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_6863_negative__zle__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int)) ).

% negative_zle_0
tff(fact_6864_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N2: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nonpos_int_cases
tff(fact_6865_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),B)),B) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_6866_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(B),aa_int_int(modulo_modulo_int(A),B)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_6867_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V739175172307565963ildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(vEBT_Leafi(fFalse,fFalse)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_6868_vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_vebt_buildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(vEBT_Leafi(fFalse,fFalse)) ).

% vebt_buildupi.simps(2)
tff(fact_6869_vebt__assn__raw_Ocases,axiom,
    ! [X: produc3625547720036274456_VEBTi] :
      ( ! [A4: bool,B4: bool,Ai: bool,Bi: bool] : X != produc6084888613844515218_VEBTi(vEBT_Leaf(A4,B4),vEBT_Leafi(Ai,Bi))
     => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] : X != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))
       => ( ! [V2: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: bool,Ve3: bool] : X != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va,Vb3),Vc3),vEBT_Leafi(Vd3,Ve3))
         => ~ ! [Vd3: bool,Ve3: bool,V2: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : X != produc6084888613844515218_VEBTi(vEBT_Leaf(Vd3,Ve3),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va,Vb3),Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_6870_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,Ai2: bool,Bi2: bool] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf(A,B)),vEBT_Leafi(Ai2,Bi2)) = aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_bool_bool(fequal_bool(Ai2),A)),aa_bool_bool(fequal_bool(Bi2),B))) ).

% vebt_assn_raw.simps(1)
tff(fact_6871_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6872_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6873_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6874_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6875_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6876_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6877_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6878_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6879_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_6880_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_6881_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_6882_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_6883_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6884_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6885_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6886_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6887_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6888_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6889_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6890_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6891_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6892_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = B ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_6893_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = B ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_6894_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = B ) )
      & ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_6895_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6896_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6897_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6898_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_6899_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_6900_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(A,Z),B) = uminus_uminus_rat(B) ) )
      & ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(A,Z),B) = divide_divide_rat(minus_minus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_6901_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_6902_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_6903_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = uminus_uminus_rat(B) ) )
      & ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_6904_even__minus,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)) ) ).

% even_minus
tff(fact_6905_even__minus,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),uminus1351360451143612070nteger(A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)) ) ).

% even_minus
tff(fact_6906_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6907_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6908_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
    <=> ( ( X = Y )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6909_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
    <=> ( ( X = Y )
        | ( X = uminus1351360451143612070nteger(Y) ) ) ) ).

% power2_eq_iff
tff(fact_6910_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))) )
    <=> ( ( X = Y )
        | ( X = uminus_uminus_rat(Y) ) ) ) ).

% power2_eq_iff
tff(fact_6911_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).

% int_cases3
tff(fact_6912_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).

% not_zle_0_negative
tff(fact_6913_negative__zless__0,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_6914_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6915_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_6916_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_6917_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_6918_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : archim2889992004027027881ng_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_6919_vebt__minti_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ! [A4: bool,B4: bool] : X != vEBT_Leafi(A4,B4)
     => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : X != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : X != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) ) ) ).

% vebt_minti.cases
tff(fact_6920_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6921_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6922_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6923_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6924_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6925_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6926_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6927_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6928_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_6929_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_6930_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_6931_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_6932_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6933_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6934_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6935_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6936_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6937_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_6938_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6939_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6940_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6941_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_6942_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).

% uminus_power_if
tff(fact_6943_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_6944_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_6945_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),N)) ) ) ) ).

% uminus_power_if
tff(fact_6946_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6947_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6948_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6949_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6950_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6951_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% neg_int_cases
tff(fact_6952_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_6953_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(minus_minus_int(B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_6954_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_6955_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_6956_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),product_Pair_int_int(Q2,R2)))
     => ( ( B != zero_zero_int )
       => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),product_Pair_int_int(aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),aa_int_int(uminus_uminus_int,Q2)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,Q2)),one_one_int)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),zero_zero_int),aa_int_int(minus_minus_int(B),R2))))) ) ) ).

% zminus1_lemma
tff(fact_6957_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6958_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6959_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6960_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6961_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_6962_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_6963_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_6964_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_6965_minus__power__mult__self,axiom,
    ! [A: int,N: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_6966_minus__power__mult__self,axiom,
    ! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_6967_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_6968_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_6969_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_6970_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% minus_one_power_iff
tff(fact_6971_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% minus_one_power_iff
tff(fact_6972_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% minus_one_power_iff
tff(fact_6973_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N) = one_one_Code_integer ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),N) = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% minus_one_power_iff
tff(fact_6974_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ) ).

% minus_one_power_iff
tff(fact_6975_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_6976_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6977_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6978_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6979_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_6980_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_6981_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_6982_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_6983_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% power_minus1_odd
tff(fact_6984_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_6985_int__bit__induct,axiom,
    ! [P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(P,zero_zero_int))
     => ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K2: int] :
              ( pp(aa_int_bool(P,K2))
             => ( ( K2 != zero_zero_int )
               => pp(aa_int_bool(P,aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,bit0(one))))) ) )
         => ( ! [K2: int] :
                ( pp(aa_int_bool(P,K2))
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(P,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,bit0(one)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_6986_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6987_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6988_m1mod2k,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),one_one_int) ).

% m1mod2k
tff(fact_6989_vebt__minti_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_minti(vEBT_Leafi(A,B)) = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
      & ( ~ pp(A)
       => ( ( pp(B)
           => ( vEBT_vebt_minti(vEBT_Leafi(A,B)) = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
          & ( ~ pp(B)
           => ( vEBT_vebt_minti(vEBT_Leafi(A,B)) = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) ).

% vebt_minti.simps(1)
tff(fact_6990_vebt__maxti_Osimps_I1_J,axiom,
    ! [B: bool,A: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_maxti(vEBT_Leafi(A,B)) = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_maxti(vEBT_Leafi(A,B)) = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_maxti(vEBT_Leafi(A,B)) = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) ).

% vebt_maxti.simps(1)
tff(fact_6991_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),A))) ) ).

% sb_dec_lem'
tff(fact_6992_m1mod22k,axiom,
    ! [N: nat] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),one_one_int) ).

% m1mod22k
tff(fact_6993_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K))))) ) ).

% sb_inc_lem'
tff(fact_6994_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_6995_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),A)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),A))) ) ).

% sb_dec_lem
tff(fact_6996_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( N = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_6997_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) ) )
      & ( ( N != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_6998_vebt__minti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leafi(A4,B4) )
           => ~ ( ( pp(A4)
                 => ( Y = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                & ( ~ pp(A4)
                 => ( ( pp(B4)
                     => ( Y = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                    & ( ~ pp(B4)
                     => ( Y = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw)
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz)
               => ( Y != heap_T3487192422709364219on_nat(some_nat(Mi2)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_6999_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7000_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7001_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7002_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = uminus1351360451143612070nteger(aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7003_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int),aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_7004_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7005_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7006_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7007_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7008_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7009_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_7010_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ).

% and_int.simps
tff(fact_7011_and__int_Oelims,axiom,
    ! [X: int,Xa4: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa4) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa4),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa4))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa4),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa4))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa4),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_7012_and_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),A) = A ).

% and.idem
tff(fact_7013_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_7014_and_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.left_idem
tff(fact_7015_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.left_idem
tff(fact_7016_and_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),B) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.right_idem
tff(fact_7017_and_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.right_idem
tff(fact_7018_and__zero__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% and_zero_eq
tff(fact_7019_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_7020_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_7021_zero__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% zero_and_eq
tff(fact_7022_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_7023_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_7024_bit_Oconj__zero__left,axiom,
    ! [X: code_integer] : bit_se3949692690581998587nteger(zero_z3403309356797280102nteger,X) = zero_z3403309356797280102nteger ).

% bit.conj_zero_left
tff(fact_7025_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_7026_bit_Oconj__zero__right,axiom,
    ! [X: code_integer] : bit_se3949692690581998587nteger(X,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bit.conj_zero_right
tff(fact_7027_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_7028_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X = A ) ) ).

% real_add_minus_iff
tff(fact_7029_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_7030_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_7031_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_7032_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(uminus1351360451143612070nteger(one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_7033_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_7034_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_7035_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_7036_bit_Oconj__one__right,axiom,
    ! [X: code_integer] : bit_se3949692690581998587nteger(X,uminus1351360451143612070nteger(one_one_Code_integer)) = X ).

% bit.conj_one_right
tff(fact_7037_bit_Oconj__one__right,axiom,
    ! [X: int] : aa_int_int(bit_se725231765392027082nd_int(X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).

% bit.conj_one_right
tff(fact_7038_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_7039_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_7040_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_7041_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_7042_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_7043_and__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_7044_and__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_7045_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ).

% and_numerals(2)
tff(fact_7046_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_7047_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_7048_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_7049_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% less_numeral_Suc
tff(fact_7050_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_7051_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).

% le_numeral_Suc
tff(fact_7052_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_7053_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_7054_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_7055_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_7056_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_7057_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_7058_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_7059_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_7060_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_7061_and__numerals_I5_J,axiom,
    ! [X: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(bit0(X)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_7062_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_7063_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_7064_and__numerals_I1_J,axiom,
    ! [Y: num] : bit_se3949692690581998587nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(Y))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_7065_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_7066_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,bit0(Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_7067_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(X))),aa_num_int(numeral_numeral_int,bit0(Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_7068_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_num_nat(numeral_numeral_nat,bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(3)
tff(fact_7069_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_7070_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_7071_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,bit0(Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_7072_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(6)
tff(fact_7073_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_7074_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(4)
tff(fact_7075_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_7076_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_7077_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_7078_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,bit0(K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_numeral_bit0
tff(fact_7079_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_7080_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_7081_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_7082_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_7083_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se725231765392027082nd_int(K),L)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_and_eq
tff(fact_7084_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_and_eq
tff(fact_7085_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_and_eq
tff(fact_7086_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),C) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.assoc
tff(fact_7087_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.assoc
tff(fact_7088_and_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(bit_se725231765392027082nd_int(B),A) ).

% and.commute
tff(fact_7089_and_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),A) ).

% and.commute
tff(fact_7090_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(B),aa_int_int(bit_se725231765392027082nd_int(A),C)) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.left_commute
tff(fact_7091_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.left_commute
tff(fact_7092_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( ( A = uminus1351360451143612070nteger(one_one_Code_integer) )
        & ( B = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7093_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7094_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(X),Y))) ) ).

% AND_lower
tff(fact_7095_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),X)) ) ).

% AND_upper1
tff(fact_7096_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Y)) ) ).

% AND_upper2
tff(fact_7097_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_7098_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_7099_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_7100_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))) ).

% real_minus_mult_self_le
tff(fact_7101_minus__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_7102_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K)) ) ).

% and_less_eq
tff(fact_7103_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_7104_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_7105_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_7106_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_7107_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_7108_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_le_0_iff
tff(fact_7109_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_7110_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B)) ) ) ).

% even_and_iff
tff(fact_7111_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B)) ) ) ).

% even_and_iff
tff(fact_7112_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L)) ) ) ).

% even_and_iff_int
tff(fact_7113_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% and_one_eq
tff(fact_7114_and__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% and_one_eq
tff(fact_7115_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% and_one_eq
tff(fact_7116_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% one_and_eq
tff(fact_7117_one__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% one_and_eq
tff(fact_7118_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% one_and_eq
tff(fact_7119_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% realpow_square_minus_le
tff(fact_7120_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_7121_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_7122_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_7123_dbl__dec__def,axiom,
    ! [X: rat] : neg_nu3179335615603231917ec_rat(X) = minus_minus_rat(plus_plus_rat(X,X),one_one_rat) ).

% dbl_dec_def
tff(fact_7124_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_7125_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_7126_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_7127_and__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% and_int_rec
tff(fact_7128_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = zero_zero_int ) )
      & ( ~ ( ( K = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = L ) )
          & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,bit0(one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_7129_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_7130_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size(3)
tff(fact_7131_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(bit0(one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_co,X)) ) ) ) ).

% ln_series
tff(fact_7132_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_7133_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_7134_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(A2),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A2),X),X_1)) ) ) ) ).

% obtain_set_pred
tff(fact_7135_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),N))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_7136_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A),X_1))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X3)) ) ) ) ).

% succ_none_empty
tff(fact_7137_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A),X_1))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),A)) ) ) ) ).

% pred_none_empty
tff(fact_7138_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_7139_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_7140_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_7141_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_7142_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(A2),Z))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
         => ( ( A2 = B2 )
           => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A2),X),X_1)) ) ) ) ) ).

% obtain_set_succ
tff(fact_7143_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_7144_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_7145_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_7146_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_7147_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_7148_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_7149_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_7150_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_7151_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_7152_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_7153_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_7154_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_7155_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_7156_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_7157_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_7158_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_7159_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_7160_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,numeral_numeral_real(N)) = numeral_numeral_real(N) ).

% abs_numeral
tff(fact_7161_abs__numeral,axiom,
    ! [N: num] : abs_abs_rat(numeral_numeral_rat(N)) = numeral_numeral_rat(N) ).

% abs_numeral
tff(fact_7162_abs__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).

% abs_numeral
tff(fact_7163_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_7164_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
tff(fact_7165_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_7166_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_7167_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_7168_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_7169_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_7170_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_7171_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(times_times_int(A),A) ).

% abs_mult_self_eq
tff(fact_7172_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_7173_List_Ofinite__set,axiom,
    ! [Xs: list_real] : pp(aa_set_real_bool(finite_finite_real,set_real2(Xs))) ).

% List.finite_set
tff(fact_7174_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_7175_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_7176_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : finite3207457112153483333omplex(set_complex2(Xs)) ).

% List.finite_set
tff(fact_7177_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_7178_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_7179_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
tff(fact_7180_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_7181_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_7182_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_7183_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_7184_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_7185_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_7186_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_7187_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_7188_abs__minus,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus
tff(fact_7189_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% dvd_abs_iff
tff(fact_7190_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% dvd_abs_iff
tff(fact_7191_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% abs_dvd_iff
tff(fact_7192_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% abs_dvd_iff
tff(fact_7193_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_7194_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_7195_of__int__abs,axiom,
    ! [X: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X)) ).

% of_int_abs
tff(fact_7196_of__int__abs,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,X)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,X)) ).

% of_int_abs
tff(fact_7197_of__int__abs,axiom,
    ! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = abs_abs_rat(aa_int_rat(ring_1_of_int_rat,X)) ).

% of_int_abs
tff(fact_7198_abs__bool__eq,axiom,
    ! [P: bool] : aa_real_real(abs_abs_real,aa_bool_real(zero_n3304061248610475627l_real,P)) = aa_bool_real(zero_n3304061248610475627l_real,P) ).

% abs_bool_eq
tff(fact_7199_abs__bool__eq,axiom,
    ! [P: bool] : aa_int_int(abs_abs_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).

% abs_bool_eq
tff(fact_7200_abs__bool__eq,axiom,
    ! [P: bool] : abs_abs_Code_integer(zero_n356916108424825756nteger(P)) = zero_n356916108424825756nteger(P) ).

% abs_bool_eq
tff(fact_7201_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_7202_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_7203_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))) ).

% finite_atLeastLessThan
tff(fact_7204_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(tanh_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_7205_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% tanh_real_less_iff
tff(fact_7206_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_7207_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_7208_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7209_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_7210_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% abs_le_self_iff
tff(fact_7211_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% abs_le_self_iff
tff(fact_7212_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_7213_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_7214_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_7215_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_7216_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_7217_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_7218_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_7219_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),abs_abs_rat(A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_7220_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_7221_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_7222_abs__neg__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).

% abs_neg_numeral
tff(fact_7223_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(N) ).

% abs_neg_numeral
tff(fact_7224_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(N) ).

% abs_neg_numeral
tff(fact_7225_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(N) ).

% abs_neg_numeral
tff(fact_7226_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_7227_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_7228_abs__neg__one,axiom,
    abs_abs_Code_integer(uminus1351360451143612070nteger(one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_7229_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_7230_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Icc_iff
tff(fact_7231_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Icc_iff
tff(fact_7232_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_set_real_bool(finite_finite_real,set_or66887138388493659n_real(A,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Ico_iff
tff(fact_7233_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Ico_iff
tff(fact_7234_abs__power__minus,axiom,
    ! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) ).

% abs_power_minus
tff(fact_7235_abs__power__minus,axiom,
    ! [A: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).

% abs_power_minus
tff(fact_7236_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),N)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).

% abs_power_minus
tff(fact_7237_abs__power__minus,axiom,
    ! [A: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)) = abs_abs_rat(aa_nat_rat(power_power_rat(A),N)) ).

% abs_power_minus
tff(fact_7238_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% tanh_real_pos_iff
tff(fact_7239_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_7240_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_7241_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_7242_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,abs_abs_rat(B))))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7243_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7244_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,abs_abs_rat(B))),zero_zero_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7245_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7246_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_nonpos
tff(fact_7247_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_nonpos
tff(fact_7248_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_7249_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_7250_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_7251_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)))
    <=> ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7252_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),N)))
    <=> ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7253_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
    <=> ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7254_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7255_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_abs
tff(fact_7256_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_abs
tff(fact_7257_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_abs
tff(fact_7258_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(abs_abs_rat(A)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% power2_abs
tff(fact_7259_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% abs_power2
tff(fact_7260_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% abs_power2
tff(fact_7261_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% abs_power2
tff(fact_7262_abs__power2,axiom,
    ! [A: rat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% abs_power2
tff(fact_7263_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,bit0(Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_7264_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7265_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_cp(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7266_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_cq(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7267_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7268_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7269_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7270_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_num_nat(numeral_numeral_nat,W)))
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7271_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_7272_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7273_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% and_Suc_0_eq
tff(fact_7274_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Suc_0_and_eq
tff(fact_7275_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_7276_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
tff(fact_7277_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),A)) ).

% abs_minus_commute
tff(fact_7278_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7279_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7280_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
tff(fact_7281_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
tff(fact_7282_power__abs,axiom,
    ! [A: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N) ).

% power_abs
tff(fact_7283_power__abs,axiom,
    ! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N) ).

% power_abs
tff(fact_7284_power__abs,axiom,
    ! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N) ).

% power_abs
tff(fact_7285_power__abs,axiom,
    ! [A: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),N)) = aa_nat_rat(power_power_rat(abs_abs_rat(A)),N) ).

% power_abs
tff(fact_7286_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_7287_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_7288_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_7289_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_7290_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_7291_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_7292_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_7293_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_7294_abs__mult,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_mult
tff(fact_7295_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_7296_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_7297_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_7298_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_7299_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% abs_le_D1
tff(fact_7300_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% abs_le_D1
tff(fact_7301_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_7302_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_7303_finite__nat__set__iff__bounded,axiom,
    ! [N4: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
    <=> ? [M5: nat] :
        ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),M5)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_7304_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),N)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% bounded_nat_set_is_finite
tff(fact_7305_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_7306_finite__list,axiom,
    ! [A2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ? [Xs2: list_real] : set_real2(Xs2) = A2 ) ).

% finite_list
tff(fact_7307_finite__list,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_7308_finite__list,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_7309_finite__list,axiom,
    ! [A2: set_complex] :
      ( finite3207457112153483333omplex(A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_7310_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cr(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_7311_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cs(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_7312_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( finite3207457112153483333omplex(A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ct(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_7313_finite__lists__length__eq,axiom,
    ! [A2: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_cu(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_7314_finite__lists__length__eq,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cv(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_7315_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_cw(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_7316_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_cx(A2),N))) ) ).

% finite_lists_length_eq
tff(fact_7317_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A))) ).

% abs_ge_zero
tff(fact_7318_abs__ge__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))) ).

% abs_ge_zero
tff(fact_7319_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_7320_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_7321_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_7322_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_7323_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_7324_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_7325_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_7326_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_7327_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_7328_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_7329_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7330_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq
tff(fact_7331_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_7332_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_7333_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(times_times_rat(C),D))) ) ) ).

% abs_mult_less
tff(fact_7334_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(times_times_int(C),D))) ) ) ).

% abs_mult_less
tff(fact_7335_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),D))) ) ) ).

% abs_mult_less
tff(fact_7336_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7337_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq2
tff(fact_7338_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B)))) ).

% abs_triangle_ineq2
tff(fact_7339_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7340_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq3
tff(fact_7341_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B)))) ).

% abs_triangle_ineq3
tff(fact_7342_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7343_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7344_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_7345_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7346_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
tff(fact_7347_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_7348_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B)) ) ) ).

% abs_leI
tff(fact_7349_abs__leI,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).

% abs_leI
tff(fact_7350_abs__leI,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).

% abs_leI
tff(fact_7351_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B)) ) ).

% abs_le_D2
tff(fact_7352_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B)) ) ).

% abs_le_D2
tff(fact_7353_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).

% abs_le_D2
tff(fact_7354_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% abs_le_D2
tff(fact_7355_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B)) ) ) ).

% abs_le_iff
tff(fact_7356_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B)) ) ) ).

% abs_le_iff
tff(fact_7357_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_le_iff
tff(fact_7358_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_le_iff
tff(fact_7359_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_7360_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),abs_abs_rat(A))) ).

% abs_ge_minus_self
tff(fact_7361_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_7362_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_7363_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_less_iff
tff(fact_7364_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_less_iff
tff(fact_7365_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B)) ) ) ).

% abs_less_iff
tff(fact_7366_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),B)) ) ) ).

% abs_less_iff
tff(fact_7367_abs__real__def,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_real_def
tff(fact_7368_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B))) ) ).

% infinite_Icc
tff(fact_7369_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B))) ) ).

% infinite_Icc
tff(fact_7370_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_set_real_bool(finite_finite_real,set_or66887138388493659n_real(A,B))) ) ).

% infinite_Ico
tff(fact_7371_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_set_rat_bool(finite_finite_rat,set_or4029947393144176647an_rat(A,B))) ) ).

% infinite_Ico
tff(fact_7372_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_cy(A2),N))) ) ).

% finite_lists_length_le
tff(fact_7373_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( finite3207457112153483333omplex(A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_cz(A2),N))) ) ).

% finite_lists_length_le
tff(fact_7374_finite__lists__length__le,axiom,
    ! [A2: set_real,N: nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => finite306553202115118035t_real(collect_list_real(aa_nat4570152432052498854l_bool(aTP_Lamm_da(A2),N))) ) ).

% finite_lists_length_le
tff(fact_7375_finite__lists__length__le,axiom,
    ! [A2: set_bool,N: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_db(A2),N))) ) ).

% finite_lists_length_le
tff(fact_7376_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_dc(A2),N))) ) ).

% finite_lists_length_le
tff(fact_7377_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_dd(A2),N))) ) ).

% finite_lists_length_le
tff(fact_7378_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_7379_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X))) ).

% tanh_real_gt_neg1
tff(fact_7380_filter__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,P: fun_Pr803113274115125473t_bool] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_de(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun8917503761599146496t_bool(aTP_Lamm_df(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7381_filter__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_dg(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_dh(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_7382_filter__preserves__multiset,axiom,
    ! [M8: fun_int_nat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_di(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_dj(M8),P)))) ) ).

% filter_preserves_multiset
tff(fact_7383_filter__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,P: fun_complex_bool] :
      ( finite3207457112153483333omplex(collect_complex(aTP_Lamm_dk(M8)))
     => finite3207457112153483333omplex(collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_dl(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_7384_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),E2)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_7385_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E2)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7386_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_7387_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => ( abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_7388_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_7389_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_7390_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7391_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( aa_rat_rat(times_times_rat(abs_abs_rat(Y)),X) = abs_abs_rat(aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7392_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7393_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_int(times_times_int(aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(times_times_int(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_7394_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_7395_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_7396_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_7397_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_7398_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
        & ( ( B = A )
          | ( B = uminus1351360451143612070nteger(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7399_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = abs_abs_rat(B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = uminus_uminus_rat(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7400_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7401_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7402_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
        & ( ( A = B )
          | ( A = uminus1351360451143612070nteger(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7403_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat(A) = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
        & ( ( A = B )
          | ( A = uminus_uminus_rat(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7404_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7405_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7406_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N))) ).

% zero_le_power_abs
tff(fact_7407_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),N))) ).

% zero_le_power_abs
tff(fact_7408_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N))) ).

% zero_le_power_abs
tff(fact_7409_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).

% zero_le_power_abs
tff(fact_7410_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% abs_div_pos
tff(fact_7411_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7412_abs__if__raw,axiom,
    ! [X3: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X3) = aa_int_int(uminus_uminus_int,X3) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X3) = X3 ) ) ) ).

% abs_if_raw
tff(fact_7413_abs__if__raw,axiom,
    ! [X3: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X3) = aa_real_real(uminus_uminus_real,X3) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X3) = X3 ) ) ) ).

% abs_if_raw
tff(fact_7414_abs__if__raw,axiom,
    ! [X3: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X3) = uminus1351360451143612070nteger(X3) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X3) = X3 ) ) ) ).

% abs_if_raw
tff(fact_7415_abs__if__raw,axiom,
    ! [X3: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),zero_zero_rat))
       => ( abs_abs_rat(X3) = uminus_uminus_rat(X3) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),zero_zero_rat))
       => ( abs_abs_rat(X3) = X3 ) ) ) ).

% abs_if_raw
tff(fact_7416_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_7417_abs__if,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_if
tff(fact_7418_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = A ) ) ) ).

% abs_if
tff(fact_7419_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( abs_abs_rat(A) = A ) ) ) ).

% abs_if
tff(fact_7420_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_7421_abs__of__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_7422_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_neg
tff(fact_7423_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_neg
tff(fact_7424_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7425_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7426_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),aa_int_int(plus_plus_int(C),D)))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),C))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_7427_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(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
tff(fact_7428_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq4
tff(fact_7429_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_7430_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(X,A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(A,R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7431_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7432_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(minus_minus_int(A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(plus_plus_int(A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_7433_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7434_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(X,A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(A,R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7435_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(plus_plus_int(A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7436_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_less_iff
tff(fact_7437_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_7438_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),U)),Y))),V)) ) ) ).

% sin_bound_lemma
tff(fact_7439_add__mset__in__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,A: product_prod_int_int] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_de(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_Pro1534211998248933470t_bool(aTP_Lamm_dm(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7440_add__mset__in__multiset,axiom,
    ! [M8: fun_nat_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_dg(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dn(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_7441_add__mset__in__multiset,axiom,
    ! [M8: fun_int_nat,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_di(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_do(M8),A)))) ) ).

% add_mset_in_multiset
tff(fact_7442_add__mset__in__multiset,axiom,
    ! [M8: fun_complex_nat,A: complex] :
      ( finite3207457112153483333omplex(collect_complex(aTP_Lamm_dk(M8)))
     => finite3207457112153483333omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dp(M8),A))) ) ).

% add_mset_in_multiset
tff(fact_7443_diff__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,N4: fun_Pr5160692531401422620nt_nat] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_de(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5604724294258851521t_bool(aTP_Lamm_dq(M8),N4))) ) ).

% diff_preserves_multiset
tff(fact_7444_diff__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,N4: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aTP_Lamm_dg(M8))))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1783405840085286277t_bool(aTP_Lamm_dr(M8),N4)))) ) ).

% diff_preserves_multiset
tff(fact_7445_diff__preserves__multiset,axiom,
    ! [M8: fun_int_nat,N4: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aTP_Lamm_di(M8))))
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun5627573752281863885t_bool(aTP_Lamm_ds(M8),N4)))) ) ).

% diff_preserves_multiset
tff(fact_7446_diff__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,N4: fun_complex_nat] :
      ( finite3207457112153483333omplex(collect_complex(aTP_Lamm_dk(M8)))
     => finite3207457112153483333omplex(collect_complex(aa_fun4400319914713427657x_bool(aTP_Lamm_dt(M8),N4))) ) ).

% diff_preserves_multiset
tff(fact_7447_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,M)))) ) ).

% finite_divisors_nat
tff(fact_7448_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_7449_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
     => pp(aa_set_nat_bool(finite_finite_nat,N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_7450_abs__add__one__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_7451_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,abs_abs_rat(X)))) ).

% abs_add_one_gt_zero
tff(fact_7452_abs__add__one__gt__zero,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_7453_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X)))) ).

% abs_add_one_gt_zero
tff(fact_7454_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% of_int_leD
tff(fact_7455_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% of_int_leD
tff(fact_7456_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).

% of_int_leD
tff(fact_7457_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% of_int_lessD
tff(fact_7458_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% of_int_lessD
tff(fact_7459_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).

% of_int_lessD
tff(fact_7460_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
     => ( ( N = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).

% of_int_lessD
tff(fact_7461_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_7462_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_nat_fun_real_bool(aTP_Lamm_du,N)))) ) ).

% finite_roots_unity
tff(fact_7463_finite__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
     => finite3207457112153483333omplex(collect_complex(aTP_Lamm_dv(N))) ) ).

% finite_roots_unity
tff(fact_7464_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_le_square_iff
tff(fact_7465_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),abs_abs_rat(Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_le_square_iff
tff(fact_7466_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_le_square_iff
tff(fact_7467_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_le_square_iff
tff(fact_7468_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,X) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_7469_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_7470_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_7471_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))) = one_one_rat )
    <=> ( abs_abs_rat(X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_7472_power__even__abs,axiom,
    ! [N: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).

% power_even_abs
tff(fact_7473_power__even__abs,axiom,
    ! [N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_even_abs
tff(fact_7474_power__even__abs,axiom,
    ! [N: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).

% power_even_abs
tff(fact_7475_power__even__abs,axiom,
    ! [N: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).

% power_even_abs
tff(fact_7476_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7477_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7478_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7479_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_7480_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_le_1
tff(fact_7481_abs__square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_7482_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_7483_abs__square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_7484_abs__square__less__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_7485_abs__square__less__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_7486_abs__square__less__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_7487_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_less_1
tff(fact_7488_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).

% power_mono_even
tff(fact_7489_power__mono__even,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),abs_abs_rat(B)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).

% power_mono_even
tff(fact_7490_power__mono__even,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).

% power_mono_even
tff(fact_7491_power__mono__even,axiom,
    ! [N: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).

% power_mono_even
tff(fact_7492_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_7493_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% and_nat_rec
tff(fact_7494_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_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
tff(fact_7495_of__int__round__abs__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))) ).

% of_int_round_abs_le
tff(fact_7496_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,N)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = N ) ) ).

% round_unique'
tff(fact_7497_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,N)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one)))))
     => ( archim7778729529865785530nd_rat(X) = N ) ) ).

% round_unique'
tff(fact_7498_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7499_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_7500_suminf__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7501_suminf__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7502_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_sqrt_wlog
tff(fact_7503_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X2: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,abs_abs_rat(X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_sqrt_wlog
tff(fact_7504_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_sqrt_wlog
tff(fact_7505_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X2))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X2),aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,bit0(one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% abs_sqrt_wlog
tff(fact_7506_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,K)))) ).

% finite_Collect_less_nat
tff(fact_7507_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_dx) = zero_zero_complex ).

% suminf_zero
tff(fact_7508_suminf__zero,axiom,
    suminf_real(aTP_Lamm_dy) = zero_zero_real ).

% suminf_zero
tff(fact_7509_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_am) = zero_zero_nat ).

% suminf_zero
tff(fact_7510_suminf__zero,axiom,
    suminf_int(aTP_Lamm_dz) = zero_zero_int ).

% suminf_zero
tff(fact_7511_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size_gen(1)
tff(fact_7512_finite__interval__int1,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ea(A),B)))) ).

% finite_interval_int1
tff(fact_7513_finite__interval__int4,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_eb(A),B)))) ).

% finite_interval_int4
tff(fact_7514_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => finite3207457112153483333omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ec(N),C))) ) ).

% finite_nth_roots
tff(fact_7515_finite__interval__int2,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ed(A),B)))) ).

% finite_interval_int2
tff(fact_7516_finite__interval__int3,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ee(A),B)))) ).

% finite_interval_int3
tff(fact_7517_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_7518_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_7519_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_7520_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7521_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( finite306553202115118035t_real(M8)
     => ? [N2: nat] :
        ! [X3: list_real] :
          ( member_list_real(X3,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_real(X3)),N2)) ) ) ).

% finite_maxlen
tff(fact_7522_finite__maxlen,axiom,
    ! [M8: set_list_bool] :
      ( finite7866827932528546246t_bool(M8)
     => ? [N2: nat] :
        ! [X3: list_bool] :
          ( member_list_bool(X3,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X3)),N2)) ) ) ).

% finite_maxlen
tff(fact_7523_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( finite8100373058378681591st_nat(M8)
     => ? [N2: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X3)),N2)) ) ) ).

% finite_maxlen
tff(fact_7524_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( finite3922522038869484883st_int(M8)
     => ? [N2: nat] :
        ! [X3: list_int] :
          ( member_list_int(X3,M8)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X3)),N2)) ) ) ).

% finite_maxlen
tff(fact_7525_abs__div,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_7526_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_7527_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ag,I)))) ) ).

% finite_divisors_int
tff(fact_7528_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_7529_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_7530_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_7531_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(M),N)),M))
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7532_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,K)),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),L))) ) ).

% even_abs_add_iff
tff(fact_7533_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),aa_int_int(abs_abs_int,L))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),L))) ) ).

% even_add_abs_iff
tff(fact_7534_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
           => ? [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I3))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7535_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),Z))),one_one_int)),D))),Z)) ) ).

% decr_lemma
tff(fact_7536_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(plus_plus_int(X),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),Z))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_7537_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,vEBT_Leafi(X21,X222)) = zero_zero_nat ).

% VEBTi.size_gen(2)
tff(fact_7538_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7539_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A))) ).

% complex_mod_triangle_ineq2
tff(fact_7540_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
         => ? [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N))
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7541_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_ef,X)) ) ).

% monoseq_arctan_series
tff(fact_7542_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_eg,X)) ) ) ).

% arctan_series
tff(fact_7543_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_7544_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) = suminf_real(aTP_Lamm_eh) ).

% pi_series
tff(fact_7545_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B4: real,A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( ! [X3: real] :
                    ( pp(aa_set_real_bool(member_real(X3),A7))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B4)) )
               => ( pp(aa_set_real_bool(P,A7))
                 => pp(aa_set_real_bool(P,insert_real(B4,A7))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7546_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B4: rat,A7: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A7))
             => ( ! [X3: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X3),A7))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),B4)) )
               => ( pp(aa_set_rat_bool(P,A7))
                 => pp(aa_set_rat_bool(P,insert_rat(B4,A7))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7547_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B4: num,A7: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A7))
             => ( ! [X3: num] :
                    ( pp(aa_set_num_bool(member_num(X3),A7))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),B4)) )
               => ( pp(aa_set_num_bool(P,A7))
                 => pp(aa_set_num_bool(P,insert_num(B4,A7))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7548_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B4: nat,A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( ! [X3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X3),A7))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),B4)) )
               => ( pp(aa_set_nat_bool(P,A7))
                 => pp(aa_set_nat_bool(P,insert_nat(B4,A7))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7549_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B4: int,A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( ! [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),A7))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),B4)) )
               => ( pp(aa_set_int_bool(P,A7))
                 => pp(aa_set_int_bool(P,insert_int(B4,A7))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7550_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [B4: code_integer,A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( ! [X3: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A7))
                   => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X3),B4)) )
               => ( pp(aa_set1374477347934578310r_bool(P,A7))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(B4,A7))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A2)) ) ) ) ).

% finite_linorder_max_induct
tff(fact_7551_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_7552_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(arctan,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_7553_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_arctan_iff
tff(fact_7554_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_7555_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_7556_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_arctan_iff
tff(fact_7557_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_7558_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% arctan_less_iff
tff(fact_7559_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone
tff(fact_7560_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_7561_pi__not__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).

% pi_not_less_zero
tff(fact_7562_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_7563_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_7564_arctan__ubound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ).

% arctan_ubound
tff(fact_7565_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) ).

% arctan_one
tff(fact_7566_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ).

% arctan_bounded
tff(fact_7567_arctan__lbound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y))) ).

% arctan_lbound
tff(fact_7568_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_7569_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,bit0(one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) ).

% machin_Euler
tff(fact_7570_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,bit0(one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_7571_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),numeral_numeral_real(bit0(bit0(one))))) ).

% pi_less_4
tff(fact_7572_pi__ge__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(bit0(one))),pi)) ).

% pi_ge_two
tff(fact_7573_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) != numeral_numeral_real(bit0(one)) ).

% pi_half_neq_two
tff(fact_7574_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7575_pi__half__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one)))) ).

% pi_half_less_two
tff(fact_7576_pi__half__le__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one)))) ).

% pi_half_le_two
tff(fact_7577_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% monoseq_realpow
tff(fact_7578_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ).

% pi_half_gt_zero
tff(fact_7579_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ).

% pi_half_ge_zero
tff(fact_7580_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_7581_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_7582_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_7583_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(nat_set_encode,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_7584_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( S2 != bot_bot_set_real )
       => ? [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),S2))
            & ~ ? [Xa: real] :
                  ( pp(aa_set_real_bool(member_real(Xa),S2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7585_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( pp(aa_set_rat_bool(finite_finite_rat,S2))
     => ( ( S2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( pp(aa_set_rat_bool(member_rat(X2),S2))
            & ~ ? [Xa: rat] :
                  ( pp(aa_set_rat_bool(member_rat(Xa),S2))
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7586_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( pp(aa_set_num_bool(finite_finite_num,S2))
     => ( ( S2 != bot_bot_set_num )
       => ? [X2: num] :
            ( pp(aa_set_num_bool(member_num(X2),S2))
            & ~ ? [Xa: num] :
                  ( pp(aa_set_num_bool(member_num(Xa),S2))
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7587_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( S2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),S2))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S2))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7588_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( S2 != bot_bot_set_int )
       => ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),S2))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S2))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7589_ex__min__if__finite,axiom,
    ! [S2: set_Code_integer] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,S2))
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),S2))
            & ~ ? [Xa: code_integer] :
                  ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa),S2))
                  & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Xa),X2)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7590_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),X7))
           => ? [Xa: real] :
                ( pp(aa_set_real_bool(member_real(Xa),X7))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Xa)) ) )
       => ~ pp(aa_set_real_bool(finite_finite_real,X7)) ) ) ).

% infinite_growing
tff(fact_7591_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( pp(aa_set_rat_bool(member_rat(X2),X7))
           => ? [Xa: rat] :
                ( pp(aa_set_rat_bool(member_rat(Xa),X7))
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Xa)) ) )
       => ~ pp(aa_set_rat_bool(finite_finite_rat,X7)) ) ) ).

% infinite_growing
tff(fact_7592_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X2: num] :
            ( pp(aa_set_num_bool(member_num(X2),X7))
           => ? [Xa: num] :
                ( pp(aa_set_num_bool(member_num(Xa),X7))
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Xa)) ) )
       => ~ pp(aa_set_num_bool(finite_finite_num,X7)) ) ) ).

% infinite_growing
tff(fact_7593_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),X7))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X7))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X7)) ) ) ).

% infinite_growing
tff(fact_7594_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),X7))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X7))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X7)) ) ) ).

% infinite_growing
tff(fact_7595_infinite__growing,axiom,
    ! [X7: set_Code_integer] :
      ( ( X7 != bot_bo3990330152332043303nteger )
     => ( ! [X2: code_integer] :
            ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X2),X7))
           => ? [Xa: code_integer] :
                ( pp(aa_set1374477347934578310r_bool(member_Code_integer(Xa),X7))
                & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Xa)) ) )
       => ~ pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,X7)) ) ) ).

% infinite_growing
tff(fact_7596_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ).

% arctan_double
tff(fact_7597_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( pp(aa_set_real_bool(P,bot_bot_set_real))
       => ( ! [B4: real,A7: set_real] :
              ( pp(aa_set_real_bool(finite_finite_real,A7))
             => ( ! [X3: real] :
                    ( pp(aa_set_real_bool(member_real(X3),A7))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B4),X3)) )
               => ( pp(aa_set_real_bool(P,A7))
                 => pp(aa_set_real_bool(P,insert_real(B4,A7))) ) ) )
         => pp(aa_set_real_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7598_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_bool] :
      ( pp(aa_set_rat_bool(finite_finite_rat,A2))
     => ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
       => ( ! [B4: rat,A7: set_rat] :
              ( pp(aa_set_rat_bool(finite_finite_rat,A7))
             => ( ! [X3: rat] :
                    ( pp(aa_set_rat_bool(member_rat(X3),A7))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B4),X3)) )
               => ( pp(aa_set_rat_bool(P,A7))
                 => pp(aa_set_rat_bool(P,insert_rat(B4,A7))) ) ) )
         => pp(aa_set_rat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7599_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_bool] :
      ( pp(aa_set_num_bool(finite_finite_num,A2))
     => ( pp(aa_set_num_bool(P,bot_bot_set_num))
       => ( ! [B4: num,A7: set_num] :
              ( pp(aa_set_num_bool(finite_finite_num,A7))
             => ( ! [X3: num] :
                    ( pp(aa_set_num_bool(member_num(X3),A7))
                   => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B4),X3)) )
               => ( pp(aa_set_num_bool(P,A7))
                 => pp(aa_set_num_bool(P,insert_num(B4,A7))) ) ) )
         => pp(aa_set_num_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7600_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
       => ( ! [B4: nat,A7: set_nat] :
              ( pp(aa_set_nat_bool(finite_finite_nat,A7))
             => ( ! [X3: nat] :
                    ( pp(aa_set_nat_bool(member_nat(X3),A7))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B4),X3)) )
               => ( pp(aa_set_nat_bool(P,A7))
                 => pp(aa_set_nat_bool(P,insert_nat(B4,A7))) ) ) )
         => pp(aa_set_nat_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7601_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(P,bot_bot_set_int))
       => ( ! [B4: int,A7: set_int] :
              ( pp(aa_set_int_bool(finite_finite_int,A7))
             => ( ! [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),A7))
                   => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B4),X3)) )
               => ( pp(aa_set_int_bool(P,A7))
                 => pp(aa_set_int_bool(P,insert_int(B4,A7))) ) ) )
         => pp(aa_set_int_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7602_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se2712834123562403997r_bool] :
      ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A2))
     => ( pp(aa_set1374477347934578310r_bool(P,bot_bo3990330152332043303nteger))
       => ( ! [B4: code_integer,A7: set_Code_integer] :
              ( pp(aa_set1374477347934578310r_bool(finite6017078050557962740nteger,A7))
             => ( ! [X3: code_integer] :
                    ( pp(aa_set1374477347934578310r_bool(member_Code_integer(X3),A7))
                   => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B4),X3)) )
               => ( pp(aa_set1374477347934578310r_bool(P,A7))
                 => pp(aa_set1374477347934578310r_bool(P,insert_Code_integer(B4,A7))) ) ) )
         => pp(aa_set1374477347934578310r_bool(P,A2)) ) ) ) ).

% finite_linorder_min_induct
tff(fact_7603_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),pi)),numeral_numeral_real(bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_7604_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_eg,X)) ) ).

% summable_arctan_series
tff(fact_7605_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))))),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7606_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
    <=> ! [M5: int] :
        ? [N7: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M5),aa_int_int(abs_abs_int,N7)))
          & pp(aa_set_int_bool(member_int(N7),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_7607_vebt__buildupi__refines,axiom,
    ! [N: nat] : refine5565527176597971370_VEBTi(vEBT_vebt_buildupi(N),vEBT_V739175172307565963ildupi(N)) ).

% vebt_buildupi_refines
tff(fact_7608_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F5: heap_T8145700208782473153_VEBTi,N: nat] :
      ( refine5565527176597971370_VEBTi(F,F5)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(N,F),vEBT_V1859673955506687831_VEBTi(N,F5)) ) ).

% refines_replicate
tff(fact_7609_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_7610_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_7611_summable__zero,axiom,
    summable_complex(aTP_Lamm_dx) ).

% summable_zero
tff(fact_7612_summable__zero,axiom,
    summable_real(aTP_Lamm_dy) ).

% summable_zero
tff(fact_7613_summable__zero,axiom,
    summable_nat(aTP_Lamm_am) ).

% summable_zero
tff(fact_7614_summable__zero,axiom,
    summable_int(aTP_Lamm_dz) ).

% summable_zero
tff(fact_7615_summable__single,axiom,
    ! [I: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(I),F)) ).

% summable_single
tff(fact_7616_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(I),F)) ).

% summable_single
tff(fact_7617_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ek(I),F)) ).

% summable_single
tff(fact_7618_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_el(I),F)) ).

% summable_single
tff(fact_7619_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_em(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7620_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_7621_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7622_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_7623_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_en(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7624_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7625_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ep(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7626_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7627_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_er(P),F)) ) ).

% summable_If_finite
tff(fact_7628_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_es(P),F)) ) ).

% summable_If_finite
tff(fact_7629_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_et(P),F)) ) ).

% summable_If_finite
tff(fact_7630_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_eu(P),F)) ) ).

% summable_If_finite
tff(fact_7631_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ev(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7632_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ew(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7633_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ex(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7634_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ey(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7635_cos__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi2
tff(fact_7636_cos__periodic__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi
tff(fact_7637_sin__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi2
tff(fact_7638_sin__periodic__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi
tff(fact_7639_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7640_sin__npi2,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).

% sin_npi2
tff(fact_7641_sin__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_7642_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_npi_int
tff(fact_7643_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_7644_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_7645_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7646_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7647_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7648_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) = one_one_real ).

% sin_pi_half
tff(fact_7649_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_7650_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_7651_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_7652_cos__npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi
tff(fact_7653_cos__npi2,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% cos_npi2
tff(fact_7654_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7655_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7656_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7657_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7658_sin__2npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7659_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7660_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_7661_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7662_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).

% cos_int_2pin
tff(fact_7663_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(bit0(one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7664_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(bit0(one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7665_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = one_one_real ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),N))
       => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% cos_npi_int
tff(fact_7666_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: fun_bo851793448214054487_VEBTi,F12: fun_bo851793448214054487_VEBTi,F22: fun_op5127434927081926045_VEBTi,F23: fun_op5127434927081926045_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A4: bool,B4: bool] : refine5565527176597971370_VEBTi(aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F1,A4),B4),aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F12,A4),B4))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info2),Deg2),TreeArray),Summary2),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F23,Info2),Deg2),TreeArray),Summary2))
         => refine5565527176597971370_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F1,Ti),vEBT_c6028912655521741485_VEBTi(F23,F12,Ti2)) ) ) ) ).

% refines_case_VEBTi
tff(fact_7667_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_7668_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_7669_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_7670_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R: real,A4: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(cos_real,A4)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(sin_real,A4)) ) ) ).

% polar_Ex
tff(fact_7671_sin__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_7672_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_ez(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7673_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fa,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7674_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_7675_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_7676_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fb(F),G)) ) ) ).

% summable_add
tff(fact_7677_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fc(F),G)) ) ) ).

% summable_add
tff(fact_7678_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_fd(F),G)) ) ) ).

% summable_add
tff(fact_7679_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fe(F),C)) ) ).

% summable_mult2
tff(fact_7680_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ff(F),C)) ) ).

% summable_mult
tff(fact_7681_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fg(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7682_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ep(F),C)) ) ).

% summable_divide
tff(fact_7683_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),C)) ) ).

% summable_divide
tff(fact_7684_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_em(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7685_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7686_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7687_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_7688_sin__double,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,X))),aa_complex_complex(cos_complex,X)) ).

% sin_double
tff(fact_7689_sin__double,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(sin_real,X))),aa_real_real(cos_real,X)) ).

% sin_double
tff(fact_7690_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_fi(F),Z)) ) ) ).

% powser_insidea
tff(fact_7691_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_fk(F),Z)) ) ) ).

% powser_insidea
tff(fact_7692_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_7693_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => summable_complex(F) ) ) ).

% summable_finite
tff(fact_7694_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_7695_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_7696_summable__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_7697_sin__x__le__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),X)) ) ).

% sin_x_le_x
tff(fact_7698_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_en(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7699_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7700_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7701_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7702_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_7703_cos__arctan__not__zero,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_7704_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fb(F),G)) ) ) ) ).

% suminf_add
tff(fact_7705_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fc(F),G)) ) ) ) ).

% suminf_add
tff(fact_7706_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(plus_plus_int(suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_fd(F),G)) ) ) ) ).

% suminf_add
tff(fact_7707_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_fe(F),C)) ) ) ).

% suminf_mult2
tff(fact_7708_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ff(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_7709_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ep(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_7710_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_7711_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real)) ).

% sin_cos_le1
tff(fact_7712_cos__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cos_squared_eq
tff(fact_7713_cos__squared__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cos_squared_eq
tff(fact_7714_sin__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% sin_squared_eq
tff(fact_7715_sin__squared__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% sin_squared_eq
tff(fact_7716_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_7717_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_7718_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_7719_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N7: nat] : aa_nat_real(F,N7) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7720_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N7: nat] : aa_nat_nat(F,N7) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7721_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N7: nat] : aa_nat_int(F,N7) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7722_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_7723_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_7724_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_7725_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_cp(F)) ).

% summable_zero_power'
tff(fact_7726_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_cq(F)) ).

% summable_zero_power'
tff(fact_7727_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_fl(F)) ).

% summable_zero_power'
tff(fact_7728_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_cp(F)) ).

% summable_0_powser
tff(fact_7729_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_cq(F)) ).

% summable_0_powser
tff(fact_7730_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero
tff(fact_7731_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_7732_sin__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_ge_zero
tff(fact_7733_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fm(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7734_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7735_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fm(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7736_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7737_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_7738_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_7739_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_7740_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_fo(F),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7741_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,M: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fp(F),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7742_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(bit0(one))))) ).

% cos_diff_cos
tff(fact_7743_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)),numeral_numeral_real(bit0(one))))) ).

% cos_diff_cos
tff(fact_7744_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(bit0(one))))) ).

% sin_diff_sin
tff(fact_7745_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(bit0(one))))) ).

% sin_diff_sin
tff(fact_7746_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(bit0(one))))) ).

% sin_plus_sin
tff(fact_7747_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(bit0(one))))) ).

% sin_plus_sin
tff(fact_7748_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(bit0(one))) ).

% cos_times_sin
tff(fact_7749_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(bit0(one))) ).

% cos_times_sin
tff(fact_7750_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(bit0(one))) ).

% sin_times_cos
tff(fact_7751_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(bit0(one))) ).

% sin_times_cos
tff(fact_7752_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(bit0(one))) ).

% sin_times_sin
tff(fact_7753_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(bit0(one))) ).

% sin_times_sin
tff(fact_7754_cos__double,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cos_double
tff(fact_7755_cos__double,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cos_double
tff(fact_7756_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% cos_double_sin
tff(fact_7757_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% cos_double_sin
tff(fact_7758_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I2))) ) ) ) ).

% suminf_pos_iff
tff(fact_7759_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I2))) ) ) ) ).

% suminf_pos_iff
tff(fact_7760_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I2))) ) ) ) ).

% suminf_pos_iff
tff(fact_7761_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_7762_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_7763_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_7764_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(bit0(one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7765_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z)) ) ) ).

% powser_inside
tff(fact_7766_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z)) ) ) ).

% powser_inside
tff(fact_7767_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_7768_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_7769_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_7770_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_fg(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7771_summable__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_7772_summable__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_7773_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real))
     => summable_real(aa_real_fun_nat_real(power_power_real,X)) ) ).

% complete_algebra_summable_geometric
tff(fact_7774_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real))
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_7775_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),pi))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_7776_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_7777_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I2: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7778_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real )
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),pi))
            & ( X = aa_real_real(cos_real,T3) )
            & ( Y = aa_real_real(sin_real,T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_7779_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one))))) ).

% sin_expansion_lemma
tff(fact_7780_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one)))))) ).

% cos_expansion_lemma
tff(fact_7781_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(bit0(one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_7782_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(bit0(one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_7783_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(bit0(one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_7784_cos__is__zero,axiom,
    ? [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),numeral_numeral_real(bit0(one))))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X2 ) ) ) ).

% cos_is_zero
tff(fact_7785_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_7786_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),pi))
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),pi))
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_7787_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real )
         => ? [T3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
              & ( X = aa_real_real(cos_real,T3) )
              & ( Y = aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7788_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real )
     => ? [T3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)))
          & ( X = aa_real_real(cos_real,T3) )
          & ( Y = aa_real_real(sin_real,T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7789_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real )
     => ~ ! [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)))
             => ( ( X = aa_real_real(cos_real,T3) )
               => ( Y != aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7790_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fm(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7791_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7792_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fm(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7793_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7794_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N9: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N9))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_em(F),N9)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7795_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N9: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N9))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_fq(F),N9)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_7796_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7797_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real))
     => ( ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7798_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M8: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),R0))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N2))),M8))
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_fr(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_7799_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(bit0(one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(bit0(one))))) ).

% cos_plus_cos
tff(fact_7800_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(bit0(one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(bit0(one))))) ).

% cos_plus_cos
tff(fact_7801_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(bit0(one))) ).

% cos_times_cos
tff(fact_7802_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(bit0(one))) ).

% cos_times_cos
tff(fact_7803_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_7804_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_7805_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero2
tff(fact_7806_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_7807_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))) ).

% sin_30
tff(fact_7808_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(bit0(one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_7809_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero
tff(fact_7810_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7811_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7812_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7813_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))) ).

% cos_60
tff(fact_7814_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X5: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X5)),numeral_numeral_real(bit0(one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7815_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7816_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_real) ).

% cos_double_cos
tff(fact_7817_cos__treble__cos,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(bit0(one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X))) ).

% cos_treble_cos
tff(fact_7818_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_7819_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M3))
         => ? [N9: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N9))
              & pp(aa_set_nat_bool(member_nat(N9),S2)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).

% unbounded_k_infinite
tff(fact_7820_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
    <=> ! [M5: nat] :
        ? [N7: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N7))
          & pp(aa_set_nat_bool(member_nat(N7),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_7821_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_7822_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_7823_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7824_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_7825_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_7826_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_7827_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_ge_zero
tff(fact_7828_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X5: nat] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X5)),numeral_numeral_real(bit0(one)))),pi)
        | ? [X5: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X5)),numeral_numeral_real(bit0(one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7829_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7830_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),I2))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7831_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I2: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),I2))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7832_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7833_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N7: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) )
        | ? [N7: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7834_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7835_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N7: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) )
        | ? [N7: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N7))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N7)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7836_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% tan_double
tff(fact_7837_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(tan_real,X))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% tan_double
tff(fact_7838_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T3)),aa_real_real(sin_real,T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7839_and__int_Opelims,axiom,
    ! [X: int,Xa4: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa4) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa4)))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa4),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa4))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa4),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa4))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa4),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa4))) ) ) ) ).

% and_int.pelims
tff(fact_7840_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K,L)))
     => ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_7841_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_7842_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBTi_bool,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBTi_bool(P,X))
       => pp(aa_VEBT_VEBTi_bool(P,nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,X),I))) ) ) ).

% intind
tff(fact_7843_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_nat_bool(P,X))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X)),I))) ) ) ).

% intind
tff(fact_7844_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_bool,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_int_bool(P,X))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(replicate_int(N,X)),I))) ) ) ).

% intind
tff(fact_7845_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( pp(aa_VEBT_VEBT_bool(P,X))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I))) ) ) ).

% intind
tff(fact_7846_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_Time_Heap_bool,A2: fun_VE1323945509527881313l_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_2774773592644289642e_bool(Q,X,aa_VEB2879522196611654106l_assn(aa_fun6755658226996116224l_assn(aTP_Lamm_fs(Q),A2),Y))
     => hoare_9026296798553663728t_bool(Q,vEBT_V6044891547498605301i_bool(N,X),aa_nat2784677144853144906l_assn(aa_VEB3184745993655927913l_assn(aa_fun7303525306808986255l_assn(aTP_Lamm_ft(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_7847_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_T2636463487746394924on_nat,A2: fun_VE6334927249922962060t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_7629718768684598413on_nat(Q,X,aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_fu(Q),A2),Y))
     => hoare_6480275734082232733on_nat(Q,vEBT_V792416675989592002on_nat(N,X),aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_fv(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_7848_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_Time_Heap_nat,A2: fun_VE4931415397359990460t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_3067605981109127869le_nat(Q,X,aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_fw(Q),A2),Y))
     => hoare_7964568885773372237st_nat(Q,vEBT_V7726092123322077554ei_nat(N,X),aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_fx(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_7849_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_T8145700208782473153_VEBTi,A2: fun_VE905575822219544865i_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_1429296392585015714_VEBTi(Q,X,aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_fy(Q),A2),Y))
     => hoare_3904069481286416050_VEBTi(Q,vEBT_V1859673955506687831_VEBTi(N,X),aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_fz(Q),A2),Y),N)) ) ).

% repli_cons_repl
tff(fact_7850_repli__emp,axiom,
    ! [X: heap_Time_Heap_bool,A2: fun_VE1323945509527881313l_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_2774773592644289642e_bool(one_one_assn,X,aa_VEB2879522196611654106l_assn(A2,Y))
     => hoare_9026296798553663728t_bool(one_one_assn,vEBT_V6044891547498605301i_bool(N,X),vEBT_L5320321625355440422T_bool(A2,replicate_VEBT_VEBT(N,Y))) ) ).

% repli_emp
tff(fact_7851_repli__emp,axiom,
    ! [X: heap_T2636463487746394924on_nat,A2: fun_VE6334927249922962060t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_7629718768684598413on_nat(one_one_assn,X,aa_VEB931861714549014589t_assn(A2,Y))
     => hoare_6480275734082232733on_nat(one_one_assn,vEBT_V792416675989592002on_nat(N,X),vEBT_L8010285020845282001on_nat(A2,replicate_VEBT_VEBT(N,Y))) ) ).

% repli_emp
tff(fact_7852_repli__emp,axiom,
    ! [X: heap_Time_Heap_nat,A2: fun_VE4931415397359990460t_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_3067605981109127869le_nat(one_one_assn,X,aa_VEB8640158415945897581t_assn(A2,Y))
     => hoare_7964568885773372237st_nat(one_one_assn,vEBT_V7726092123322077554ei_nat(N,X),vEBT_L8296926524756676353BT_nat(A2,replicate_VEBT_VEBT(N,Y))) ) ).

% repli_emp
tff(fact_7853_repli__emp,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,A2: fun_VE905575822219544865i_assn,Y: vEBT_VEBT,N: nat] :
      ( hoare_1429296392585015714_VEBTi(one_one_assn,X,aa_VEB3717855479479287634i_assn(A2,Y))
     => hoare_3904069481286416050_VEBTi(one_one_assn,vEBT_V1859673955506687831_VEBTi(N,X),vEBT_L6296928887356842470_VEBTi(A2,replicate_VEBT_VEBT(N,Y))) ) ).

% repli_emp
tff(fact_7854_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_7855_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_7856_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7857_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_7858_tan__periodic__pi,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(tan_real,X) ).

% tan_periodic_pi
tff(fact_7859_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X)) = N ).

% length_replicate
tff(fact_7860_length__replicate,axiom,
    ! [N: nat,X: real] : size_size_list_real(replicate_real(N,X)) = N ).

% length_replicate
tff(fact_7861_length__replicate,axiom,
    ! [N: nat,X: bool] : size_size_list_bool(replicate_bool(N,X)) = N ).

% length_replicate
tff(fact_7862_length__replicate,axiom,
    ! [N: nat,X: nat] : size_size_list_nat(replicate_nat(N,X)) = N ).

% length_replicate
tff(fact_7863_length__replicate,axiom,
    ! [N: nat,X: int] : size_size_list_int(replicate_int(N,X)) = N ).

% length_replicate
tff(fact_7864_map__replicate,axiom,
    ! [F: fun_nat_bool,N: nat,X: nat] : map_nat_bool(F,replicate_nat(N,X)) = replicate_bool(N,aa_nat_bool(F,X)) ).

% map_replicate
tff(fact_7865_map__replicate,axiom,
    ! [F: fun_nat_nat,N: nat,X: nat] : map_nat_nat(F,replicate_nat(N,X)) = replicate_nat(N,aa_nat_nat(F,X)) ).

% map_replicate
tff(fact_7866_map__replicate,axiom,
    ! [F: fun_VEBT_VEBT_real,N: nat,X: vEBT_VEBT] : map_VEBT_VEBT_real(F,replicate_VEBT_VEBT(N,X)) = replicate_real(N,aa_VEBT_VEBT_real(F,X)) ).

% map_replicate
tff(fact_7867_map__replicate,axiom,
    ! [F: fun_VEBT_VEBT_nat,N: nat,X: vEBT_VEBT] : map_VEBT_VEBT_nat(F,replicate_VEBT_VEBT(N,X)) = replicate_nat(N,aa_VEBT_VEBT_nat(F,X)) ).

% map_replicate
tff(fact_7868_map__replicate,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,N: nat,X: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,replicate_VEBT_VEBT(N,X)) = replicate_VEBT_VEBT(N,aa_VEB5832953974749077948T_VEBT(F,X)) ).

% map_replicate
tff(fact_7869_Ball__set__replicate,axiom,
    ! [N: nat,A: real,P: fun_real_bool] :
      ( ! [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(replicate_real(N,A))))
         => pp(aa_real_bool(P,X5)) )
    <=> ( pp(aa_real_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7870_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
         => pp(aa_nat_bool(P,X5)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7871_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(replicate_int(N,A))))
         => pp(aa_int_bool(P,X5)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7872_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A)))
         => pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_7873_Bex__set__replicate,axiom,
    ! [N: nat,A: real,P: fun_real_bool] :
      ( ? [X5: real] :
          ( pp(aa_set_real_bool(member_real(X5),set_real2(replicate_real(N,A))))
          & pp(aa_real_bool(P,X5)) )
    <=> ( pp(aa_real_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7874_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: fun_nat_bool] :
      ( ? [X5: nat] :
          ( pp(aa_set_nat_bool(member_nat(X5),set_nat2(replicate_nat(N,A))))
          & pp(aa_nat_bool(P,X5)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7875_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: fun_int_bool] :
      ( ? [X5: int] :
          ( pp(aa_set_int_bool(member_int(X5),set_int2(replicate_int(N,A))))
          & pp(aa_int_bool(P,X5)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7876_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X5: vEBT_VEBT] :
          ( member_VEBT_VEBT(X5,set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A)))
          & pp(aa_VEBT_VEBT_bool(P,X5)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_7877_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( member_complex(X,set_complex2(replicate_complex(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7878_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( pp(aa_set_real_bool(member_real(X),set_real2(replicate_real(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7879_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7880_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7881_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_7882_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBTi] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,X),I) = X ) ) ).

% nth_replicate
tff(fact_7883_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_nat(nth_nat(replicate_nat(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_7884_nth__replicate,axiom,
    ! [I: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( aa_nat_int(nth_int(replicate_int(N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_7885_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I) = X ) ) ).

% nth_replicate
tff(fact_7886_tan__npi,axiom,
    ! [N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7887_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_7888_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_7889_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_7890_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_7891_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_7892_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_7893_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_7894_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_7895_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo851793448214054487_VEBTi,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op5127434927081926045_VEBTi,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A4: bool,B4: bool] :
          ( ( Ti = vEBT_Leafi(A4,B4) )
         => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(aa_boo772418440604470950_VEBTi(aa_boo6676949965781373448_VEBTi(F,A4),B4)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A4),B4))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_T5737551269749752165_VEBTi(vEBT_c6028912655521741485_VEBTi(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_7896_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo5340758216624605938ap_nat,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op2425996437410981688ap_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A4: bool,B4: bool] :
          ( ( Ti = vEBT_Leafi(A4,B4) )
         => pp(aa_nat_bool(time_TBOUND_nat(aa_boo3049335359029405505ap_nat(aa_boo988204379142029347ap_nat(F,A4),B4)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A4),B4))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_TBOUND_nat(aa_VEB183921879236458153ap_nat(aa_arr6934346397072338570ap_nat(aa_nat1288005622339309855ap_nat(aa_opt6697879614838309225ap_nat(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_TBOUND_nat(vEBT_c1335663792808957512ap_nat(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_7897_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: fun_bo8430231047727935170on_nat,Bnd: fun_bo5748384255075588070ol_nat,F5: fun_op5783775701855621896on_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A4: bool,B4: bool] :
          ( ( Ti = vEBT_Leafi(A4,B4) )
         => pp(aa_nat_bool(time_T8353473612707095248on_nat(aa_boo4584856347283357585on_nat(aa_boo3438790501838819827on_nat(F,A4),B4)),aa_bool_nat(aa_bool_fun_bool_nat(Bnd,A4),B4))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary2) )
           => pp(aa_nat_bool(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F5,Info2),Deg2),TreeArray),Summary2)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary2))) )
       => pp(aa_nat_bool(time_T8353473612707095248on_nat(vEBT_c6250501799366334488on_nat(F5,F,Ti)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Ti))) ) ) ).

% TBOUND_VEBT_case
tff(fact_7898_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_7899_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_7900_replicate__eqI,axiom,
    ! [Xs: list_complex,N: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = N )
     => ( ! [Y3: complex] :
            ( member_complex(Y3,set_complex2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_complex(N,X) ) ) ) ).

% replicate_eqI
tff(fact_7901_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y3,set_VEBT_VEBT2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).

% replicate_eqI
tff(fact_7902_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y3: real] :
            ( pp(aa_set_real_bool(member_real(Y3),set_real2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(N,X) ) ) ) ).

% replicate_eqI
tff(fact_7903_replicate__eqI,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( ( size_size_list_bool(Xs) = N )
     => ( ! [Y3: bool] :
            ( member_bool(Y3,set_bool2(Xs))
           => ( pp(Y3)
            <=> pp(X) ) )
       => ( Xs = replicate_bool(N,X) ) ) ) ).

% replicate_eqI
tff(fact_7904_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( size_size_list_nat(Xs) = N )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(N,X) ) ) ) ).

% replicate_eqI
tff(fact_7905_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),set_int2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(N,X) ) ) ) ).

% replicate_eqI
tff(fact_7906_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => ( X2 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7907_replicate__length__same,axiom,
    ! [Xs: list_real,X: real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
         => ( X2 = X ) )
     => ( replicate_real(size_size_list_real(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7908_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X2: bool] :
          ( member_bool(X2,set_bool2(Xs))
         => ( pp(X2)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7909_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
         => ( X2 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7910_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
         => ( X2 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7911_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_7912_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_7913_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = numeral_numeral_real(W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_7914_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_real(aTP_Lamm_ga(K),Lst) = replicate_real(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_7915_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_nat(aTP_Lamm_gb(K),Lst) = replicate_nat(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_7916_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] : map_real_VEBT_VEBT(aTP_Lamm_gc(K),Lst) = replicate_VEBT_VEBT(size_size_list_real(Lst),K) ).

% map_replicate_const
tff(fact_7917_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_bool] : map_bool_VEBT_VEBT(aTP_Lamm_gd(K),Lst) = replicate_VEBT_VEBT(size_size_list_bool(Lst),K) ).

% map_replicate_const
tff(fact_7918_map__replicate__const,axiom,
    ! [K: bool,Lst: list_nat] : map_nat_bool(aa_bool_fun_nat_bool(aTP_Lamm_ge,K),Lst) = replicate_bool(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_7919_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gf,K),Lst) = replicate_nat(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_7920_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_gg(K),Lst) = replicate_VEBT_VEBT(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_7921_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] : map_int_VEBT_VEBT(aTP_Lamm_gh(K),Lst) = replicate_VEBT_VEBT(size_size_list_int(Lst),K) ).

% map_replicate_const
tff(fact_7922_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).

% complex_add
tff(fact_7923_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_7924_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_7925_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_7926_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_7927_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_7928_set__replicate__Suc,axiom,
    ! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_7929_set__replicate__Suc,axiom,
    ! [N: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X)) = insert_real(X,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_7930_tan__def,axiom,
    ! [X3: complex] : aa_complex_complex(tan_complex,X3) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X3)),aa_complex_complex(cos_complex,X3)) ).

% tan_def
tff(fact_7931_tan__def,axiom,
    ! [X3: real] : aa_real_real(tan_real,X3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X3)),aa_real_real(cos_real,X3)) ).

% tan_def
tff(fact_7932_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% set_replicate_conv_if
tff(fact_7933_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% set_replicate_conv_if
tff(fact_7934_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).

% set_replicate_conv_if
tff(fact_7935_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ) ).

% set_replicate_conv_if
tff(fact_7936_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))) = one_one_real ).

% tan_45
tff(fact_7937_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_gt_zero
tff(fact_7938_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X2))) ) ) ).

% lemma_tan_total
tff(fact_7939_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
      & ( aa_real_real(tan_real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_7940_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_7941_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_7942_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_7943_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X2))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
      & ( aa_real_real(tan_real,X2) = Y )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X2 ) ) ) ).

% tan_total
tff(fact_7944_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7945_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),Y)) ).

% tan_inverse
tff(fact_7946_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7947_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7948_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
          & ( aa_real_real(tan_real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_7949_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_pos_pi2_le
tff(fact_7950_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_7951_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_7952_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_7953_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_7954_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_7955_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_7956_arctan,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_7957_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(A0,A1)))
     => ( ! [K2: int,L3: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L3)))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,bit0(one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_7958_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7959_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7960_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7961_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7962_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7963_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7964_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z5: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))))),Z5))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))))
          & ( aa_real_real(tan_real,Z5) = X ) ) ) ).

% tan_total_pi4
tff(fact_7965_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_7966_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),X))),one_one_complex)) ).

% tan_half
tff(fact_7967_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X))),one_one_real)) ).

% tan_half
tff(fact_7968_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A1)))
     => ( ! [I3: int,J: int] :
            ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I3,J)))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I3),J))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(plus_plus_int(I3),one_one_int)),J)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_7969_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_assn,Y: fun_real_assn] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun159070158279343989l_bool(aTP_Lamm_gi(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun159070158279343989l_bool(aTP_Lamm_gi(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun159070158279343989l_bool(aa_fun6385136150007740861l_bool(aTP_Lamm_gj(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7970_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_assn,Y: fun_VEBT_VEBT_assn] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun437439898845168857T_bool(aTP_Lamm_gk(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun437439898845168857T_bool(aTP_Lamm_gk(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun437439898845168857T_bool(aa_fun2621268342929013301T_bool(aTP_Lamm_gl(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7971_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_assn,Y: fun_nat_assn] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1272683333798124589t_bool(aTP_Lamm_gm(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1272683333798124589t_bool(aTP_Lamm_gm(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1272683333798124589t_bool(aa_fun7103831392407232081t_bool(aTP_Lamm_gn(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7972_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_assn,Y: fun_int_assn] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1414614411059211637t_bool(aTP_Lamm_go(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1414614411059211637t_bool(aTP_Lamm_go(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1414614411059211637t_bool(aa_fun5523883950577104445t_bool(aTP_Lamm_gp(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7973_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_assn,Y: fun_complex_assn] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun1925668587588942193x_bool(aTP_Lamm_gq(I5),X)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun1925668587588942193x_bool(aTP_Lamm_gq(I5),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun1925668587588942193x_bool(aa_fun8862831692315842871x_bool(aTP_Lamm_gr(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7974_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_gs(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_gs(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_gt(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7975_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_gu(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_gu(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_gv(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7976_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_gw(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_gw(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_gx(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7977_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_gy(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_gy(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_gz(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_7978_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ha(I5),X)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_ha(I5),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_hb(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_7979_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_7980_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_hc(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_hc(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_hd(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7981_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_he(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aTP_Lamm_he(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_hf(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7982_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_hg(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_hg(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_hh(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7983_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_hi(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_hi(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_hj(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7984_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_hk(I5),X)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_hk(I5),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_hl(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7985_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_hm(I5),X))))
     => ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_hm(I5),Y))))
       => pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_hn(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7986_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_ho(I5),X)))
     => ( finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aTP_Lamm_ho(I5),Y)))
       => finite5795047828879050333T_VEBT(collect_VEBT_VEBT(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_hp(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7987_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_hq(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_hq(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_hr(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7988_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_hs(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_hs(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ht(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_7989_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_hu(I5),X)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_hu(I5),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_hv(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_7990_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),minus_minus_rat(one_one_rat,X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7991_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7992_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( ( ( X = one_one_real )
           => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
          & ( ( X != one_one_real )
           => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_7993_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_7994_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z) = zero_zero_real )
    <=> ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_7995_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7996_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_ai,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7997_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_dy,A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_7998_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_an,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_7999_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_hw(F),A2) ).

% of_nat_sum
tff(fact_8000_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_hx(F),A2) ).

% of_nat_sum
tff(fact_8001_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_hy(F),A2) ).

% of_nat_sum
tff(fact_8002_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups2073611262835488442omplex(aTP_Lamm_hz(F),A2) ).

% of_nat_sum
tff(fact_8003_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ia(F)),A2) ).

% of_nat_sum
tff(fact_8004_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups6591440286371151544t_real(aTP_Lamm_ca(F),A2) ).

% of_nat_sum
tff(fact_8005_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_ib(F),A2) ).

% of_int_sum
tff(fact_8006_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = groups6591440286371151544t_real(aTP_Lamm_ic(F),A2) ).

% of_int_sum
tff(fact_8007_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_id(F),A2) ).

% of_int_sum
tff(fact_8008_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_ie(F),A2) ).

% of_int_sum
tff(fact_8009_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups4538972089207619220nt_int(F,A2)) = groups3049146728041665814omplex(aTP_Lamm_if(F),A2) ).

% of_int_sum
tff(fact_8010_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_ig(F),A2) ).

% of_int_sum
tff(fact_8011_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : groups2073611262835488442omplex(G,bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_8012_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_8013_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_8014_sum_Oempty,axiom,
    ! [G: fun_nat_Code_integer] : groups7501900531339628137nteger(G,bot_bot_set_nat) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_8015_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_8016_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_8017_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_8018_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_8019_sum_Oempty,axiom,
    ! [G: fun_int_Code_integer] : groups7873554091576472773nteger(G,bot_bot_set_int) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_8020_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_8021_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F2))
     => ( ( groups4541462559716669496nt_nat(F,F2) = zero_zero_nat )
      <=> ! [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),F2))
           => ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8022_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(F2)
     => ( ( groups5693394587270226106ex_nat(F,F2) = zero_zero_nat )
      <=> ! [X5: complex] :
            ( member_complex(X5,F2)
           => ( aa_complex_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8023_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F2) = zero_zero_nat )
      <=> ! [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),F2))
           => ( aa_nat_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8024_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_8025_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_8026_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8027_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ finite3207457112153483333omplex(A2)
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8028_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8029_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8030_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ finite3207457112153483333omplex(A2)
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8031_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_8032_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ finite3207457112153483333omplex(A2)
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_8033_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups3539618377306564664at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_8034_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = one_one_real ) ) ) ).

% powr_zero_eq_one
tff(fact_8035_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)))
    <=> ( X != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8036_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8037_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel_iff
tff(fact_8038_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ih(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ih(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8039_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ii(A),B),S2) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ii(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8040_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8041_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ij(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ij(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta
tff(fact_8042_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ik(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ik(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8043_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_il(A),B),S2) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_il(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8044_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_im(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_im(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8045_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( finite3207457112153483333omplex(S2)
     => ( ( member_complex(A,S2)
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_in(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ member_complex(A,S2)
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_in(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_8046_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_io(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_io(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_8047_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ip(A),B),S2) = aa_VEBT_VEBT_rat(B,A) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ip(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_8048_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_iq(A),B),S2) = aa_real_complex(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_iq(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8049_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ir(A),B),S2) = aa_VEBT_VEBT_complex(B,A) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups1794756597179926696omplex(aa_fun8982142231764154876omplex(aTP_Lamm_ir(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8050_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_is(A),B),S2) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
         => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_is(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8051_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_it(A),B),S2) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_it(A),B),S2) = zero_zero_complex ) ) ) ) ).

% sum.delta'
tff(fact_8052_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_iu(A),B),S2) = aa_real_real(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_iu(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8053_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_iv(A),B),S2) = aa_VEBT_VEBT_real(B,A) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_iv(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8054_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_iw(A),B),S2) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_iw(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8055_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( finite3207457112153483333omplex(S2)
     => ( ( member_complex(A,S2)
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ix(A),B),S2) = aa_complex_real(B,A) ) )
        & ( ~ member_complex(A,S2)
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ix(A),B),S2) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_8056_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,S2))
     => ( ( pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iy(A),B),S2) = aa_real_rat(B,A) ) )
        & ( ~ pp(aa_set_real_bool(member_real(A),S2))
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iy(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_8057_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_iz(A),B),S2) = aa_VEBT_VEBT_rat(B,A) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_iz(A),B),S2) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_8058_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8059_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ~ member_VEBT_VEBT(X,A2)
       => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8060_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8061_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ~ member_complex(X,A2)
       => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8062_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = plus_plus_rat(aa_real_rat(G,X),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8063_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ~ member_VEBT_VEBT(X,A2)
       => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8064_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
       => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = plus_plus_rat(aa_nat_rat(G,X),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8065_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ~ pp(aa_set_int_bool(member_int(X),A2))
       => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8066_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ~ member_complex(X,A2)
       => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8067_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ~ pp(aa_set_real_bool(member_real(X),A2))
       => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_8068_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8069_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_8070_powr__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X ) ) ).

% powr_one
tff(fact_8071_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_8072_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_powr_numeral_real
tff(fact_8073_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_ja(F),A2))) ).

% sum_abs_ge_zero
tff(fact_8074_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_jb(F),A2))) ).

% sum_abs_ge_zero
tff(fact_8075_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_8076_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_8077_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_complex(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8078_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)),aa_nat_rat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8079_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8080_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_Code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups7501900531339628137nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups7501900531339628137nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = plus_p5714425477246183910nteger(groups7501900531339628137nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)),aa_nat_Code_integer(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8081_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8082_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_real(G,N)) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_8083_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8084_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8085_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8086_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_Code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups7501900531339628137nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups7501900531339628137nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_p5714425477246183910nteger(groups7501900531339628137nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_Code_integer(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8087_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8088_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_8089_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N)) ) ) ).

% powr_numeral
tff(fact_8090_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_cp(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_cp(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_8091_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_jc(C),A2) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_jc(C),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_8092_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aTP_Lamm_cq(C),A2) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aTP_Lamm_cq(C),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_8093_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_jd(C),D),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_jd(C),D),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_8094_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_je(C),D),A2) = divide_divide_rat(aa_nat_rat(C,zero_zero_nat),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_je(C),D),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_8095_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jf(C),D),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jf(C),D),A2) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_8096_square__powr__half,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_8097_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fc(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_8098_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_jg(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_8099_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fb(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).

% sum.distrib
tff(fact_8100_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_jh(G),H),A2) = aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_8101_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X2)),A2))
           => ( aa_nat_nat(F,aa_nat_nat(suc,X2)) = aa_nat_nat(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8102_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X2)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X2)) = aa_nat_real(G,aa_nat_nat(suc,X2)) ) )
       => ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8103_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8104_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8105_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8106_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8107_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_8108_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups7501900531339628137nteger(F,A2))) ) ).

% sum_nonneg
tff(fact_8109_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,X2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups7713935264441627589nteger(F,A2))) ) ).

% sum_nonneg
tff(fact_8110_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,X2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups5748017345553531991nteger(F,A2))) ) ).

% sum_nonneg
tff(fact_8111_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,A2))) ) ).

% sum_nonneg
tff(fact_8112_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,X2))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups6621422865394947399nteger(F,A2))) ) ).

% sum_nonneg
tff(fact_8113_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8114_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8115_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8116_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8117_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_8118_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),zero_z3403309356797280102nteger)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,A2)),zero_z3403309356797280102nteger)) ) ).

% sum_nonpos
tff(fact_8119_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_real_Code_integer(F,X2)),zero_z3403309356797280102nteger)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7713935264441627589nteger(F,A2)),zero_z3403309356797280102nteger)) ) ).

% sum_nonpos
tff(fact_8120_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_VEB5935388676523804373nteger(F,X2)),zero_z3403309356797280102nteger)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups5748017345553531991nteger(F,A2)),zero_z3403309356797280102nteger)) ) ).

% sum_nonpos
tff(fact_8121_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X2)),zero_z3403309356797280102nteger)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A2)),zero_z3403309356797280102nteger)) ) ).

% sum_nonpos
tff(fact_8122_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F,X2)),zero_z3403309356797280102nteger)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups6621422865394947399nteger(F,A2)),zero_z3403309356797280102nteger)) ) ).

% sum_nonpos
tff(fact_8123_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups2073611262835488442omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: nat] :
            ( pp(aa_set_nat_bool(member_nat(A4),A2))
           => ( aa_nat_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8124_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A2))
           => ( aa_real_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8125_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VE5569070535936029429omplex,A2: set_VEBT_VEBT] :
      ( ( groups1794756597179926696omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: vEBT_VEBT] :
            ( member_VEBT_VEBT(A4,A2)
           => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8126_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: int] :
            ( pp(aa_set_int_bool(member_int(A4),A2))
           => ( aa_int_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8127_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A2))
           => ( aa_real_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8128_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT] :
      ( ( groups2240296850493347238T_real(G,A2) != zero_zero_real )
     => ~ ! [A4: vEBT_VEBT] :
            ( member_VEBT_VEBT(A4,A2)
           => ( aa_VEBT_VEBT_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8129_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A4: int] :
            ( pp(aa_set_int_bool(member_int(A4),A2))
           => ( aa_int_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8130_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups5808333547571424918x_real(G,A2) != zero_zero_real )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A2)
           => ( aa_complex_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8131_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( pp(aa_set_nat_bool(member_nat(A4),A2))
           => ( aa_nat_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8132_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: real] :
            ( pp(aa_set_real_bool(member_real(A4),A2))
           => ( aa_real_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8133_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( aa_nat_nat(G,X2) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_8134_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => ( aa_complex_complex(G,X2) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_8135_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
         => ( aa_nat_real(G,X2) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_8136_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X2: int] :
          ( pp(aa_set_int_bool(member_int(X2),A2))
         => ( aa_int_int(G,X2) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_8137_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ji(F),R2),A2) ).

% sum_divide_distrib
tff(fact_8138_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A2)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),R2),A2) ).

% sum_divide_distrib
tff(fact_8139_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_jk(F),G),B2)),A2) ).

% sum_product
tff(fact_8140_sum__product,axiom,
    ! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B2: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B2)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_jm(F),G),B2),A2) ).

% sum_product
tff(fact_8141_sum__product,axiom,
    ! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,B2)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_jo(F),G),B2),A2) ).

% sum_product
tff(fact_8142_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B2: set_int] : aa_int_int(times_times_int(groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_jq(F),G),B2),A2) ).

% sum_product
tff(fact_8143_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jr(F),R2)),A2) ).

% sum_distrib_right
tff(fact_8144_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_js(F),R2),A2) ).

% sum_distrib_right
tff(fact_8145_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fe(F),R2),A2) ).

% sum_distrib_right
tff(fact_8146_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R2: int] : aa_int_int(times_times_int(groups4538972089207619220nt_int(F,A2)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_jt(F),R2),A2) ).

% sum_distrib_right
tff(fact_8147_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ju(R2),F)),A2) ).

% sum_distrib_left
tff(fact_8148_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(times_times_complex(R2),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_jv(R2),F),A2) ).

% sum_distrib_left
tff(fact_8149_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),groups6591440286371151544t_real(F,A2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(R2),F),A2) ).

% sum_distrib_left
tff(fact_8150_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A2: set_int] : aa_int_int(times_times_int(R2),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_jw(R2),F),A2) ).

% sum_distrib_left
tff(fact_8151_powr__powr,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_8152_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jx(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_8153_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_jy(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_8154_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_jz(A2),P))) = groups5754745047067104278omplex(aa_fun6813637976755515871omplex(aTP_Lamm_ka(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8155_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups1794756597179926696omplex(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_kb(A2),P))) = groups1794756597179926696omplex(aa_fun6792494735270171281omplex(aTP_Lamm_kc(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8156_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_kd(A2),P))) = groups2073611262835488442omplex(aa_fun8588439219086927967omplex(aTP_Lamm_ke(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8157_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_kf(A2),P))) = groups3049146728041665814omplex(aa_fun3606824691168040799omplex(aTP_Lamm_kg(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8158_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_jz(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_kh(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8159_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_kb(A2),P))) = groups2240296850493347238T_real(aa_fun2072160908118572431T_real(aTP_Lamm_ki(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8160_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_kf(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_kj(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8161_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_kk(A2),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_kl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8162_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_jz(A2),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_km(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8163_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,P: fun_VEBT_VEBT_bool] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups136491112297645522BT_rat(G,collect_VEBT_VEBT(aa_fun4462450708475455754T_bool(aTP_Lamm_kb(A2),P))) = groups136491112297645522BT_rat(aa_fun3193099451372820155BT_rat(aTP_Lamm_kn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_8164_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8165_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_nat(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8166_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( finite3207457112153483333omplex(T)
       => ( ! [X2: complex] :
              ( member_complex(X2,T)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: complex] :
                    ( member_complex(Xa,T)
                    & ( aa_complex_nat(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8167_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_int(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8168_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8169_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( finite3207457112153483333omplex(T)
       => ( ! [X2: complex] :
              ( member_complex(X2,T)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2))) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S))
               => ? [Xa: complex] :
                    ( member_complex(Xa,T)
                    & ( aa_complex_int(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8170_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: fun_nat_rat,I: fun_nat_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(S)
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2))) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S)
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_complex(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8171_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_rat,I: fun_int_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(S)
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2))) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S)
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_int_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8172_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_rat,I: fun_complex_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(S)
     => ( finite3207457112153483333omplex(T)
       => ( ! [X2: complex] :
              ( member_complex(X2,T)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2))) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S)
               => ? [Xa: complex] :
                    ( member_complex(Xa,T)
                    & ( aa_complex_complex(I,Xa) = X2 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8173_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_Code_integer,I: fun_nat_nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),T))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(G,X2))) )
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X2 )
                    & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(G,Xa))) ) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,S)),groups7501900531339628137nteger(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_8174_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A2))
             => ( aa_real_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8175_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,A2)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
       => ( ( groups136491112297645522BT_rat(F,A2) = zero_zero_rat )
        <=> ! [X5: vEBT_VEBT] :
              ( member_VEBT_VEBT(X5,A2)
             => ( aa_VEBT_VEBT_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8176_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A2))
             => ( aa_nat_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8177_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A2))
             => ( aa_int_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8178_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X5: complex] :
              ( member_complex(X5,A2)
             => ( aa_complex_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8179_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,X2))) )
       => ( ( groups7713935264441627589nteger(F,A2) = zero_z3403309356797280102nteger )
        <=> ! [X5: real] :
              ( pp(aa_set_real_bool(member_real(X5),A2))
             => ( aa_real_Code_integer(F,X5) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8180_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,A2)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,X2))) )
       => ( ( groups5748017345553531991nteger(F,A2) = zero_z3403309356797280102nteger )
        <=> ! [X5: vEBT_VEBT] :
              ( member_VEBT_VEBT(X5,A2)
             => ( aa_VEB5935388676523804373nteger(F,X5) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8181_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X2))) )
       => ( ( groups7501900531339628137nteger(F,A2) = zero_z3403309356797280102nteger )
        <=> ! [X5: nat] :
              ( pp(aa_set_nat_bool(member_nat(X5),A2))
             => ( aa_nat_Code_integer(F,X5) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8182_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X2))) )
       => ( ( groups7873554091576472773nteger(F,A2) = zero_z3403309356797280102nteger )
        <=> ! [X5: int] :
              ( pp(aa_set_int_bool(member_int(X5),A2))
             => ( aa_int_Code_integer(F,X5) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8183_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,X2))) )
       => ( ( groups6621422865394947399nteger(F,A2) = zero_z3403309356797280102nteger )
        <=> ! [X5: complex] :
              ( member_complex(X5,A2)
             => ( aa_com4217951221296830281nteger(F,X5) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8184_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2))) )
       => ( ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8185_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8186_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2))) )
       => ( ? [X3: complex] :
              ( member_complex(X3,A2)
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8187_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X2)),aa_nat_Code_integer(G,X2))) )
       => ( ? [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A2))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(G,X3))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups7501900531339628137nteger(F,A2)),groups7501900531339628137nteger(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8188_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X2)),aa_int_Code_integer(G,X2))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X3)),aa_int_Code_integer(G,X3))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups7873554091576472773nteger(F,A2)),groups7873554091576472773nteger(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8189_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger,G: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F,X2)),aa_com4217951221296830281nteger(G,X2))) )
       => ( ? [X3: complex] :
              ( member_complex(X3,A2)
              & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_com4217951221296830281nteger(F,X3)),aa_com4217951221296830281nteger(G,X3))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups6621422865394947399nteger(F,A2)),groups6621422865394947399nteger(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8190_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X2)),aa_int_real(G,X2))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8191_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X2)),aa_complex_real(G,X2))) )
       => ( ? [X3: complex] :
              ( member_complex(X3,A2)
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8192_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2))) )
       => ( ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8193_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2))) )
       => ( ? [X3: complex] :
              ( member_complex(X3,A2)
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8194_sum_Orelated,axiom,
    ! [R3: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,aa_complex_complex(plus_plus_complex(X15),Y1)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,aa_nat_complex(H,X2)),aa_nat_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,groups2073611262835488442omplex(H,S2)),groups2073611262835488442omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8195_sum_Orelated,axiom,
    ! [R3: fun_co1378794544173401837x_bool,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,zero_zero_complex),zero_zero_complex))
     => ( ! [X15: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,X15),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,aa_complex_complex(plus_plus_complex(X15),Y1)),aa_complex_complex(plus_plus_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,aa_int_complex(H,X2)),aa_int_complex(G,X2))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R3,groups3049146728041665814omplex(H,S2)),groups3049146728041665814omplex(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8196_sum_Orelated,axiom,
    ! [R3: fun_re1255401033344146673l_bool,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_int_real(H,X2)),aa_int_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8197_sum_Orelated,axiom,
    ! [R3: fun_re1255401033344146673l_bool,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,zero_zero_real),zero_zero_real))
     => ( ! [X15: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,X15),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( finite3207457112153483333omplex(S2)
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,aa_complex_real(H,X2)),aa_complex_real(G,X2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R3,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8198_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_bool,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,plus_plus_rat(X15,Y1)),plus_plus_rat(X23,Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_nat_rat(H,X2)),aa_nat_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8199_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_bool,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,plus_plus_rat(X15,Y1)),plus_plus_rat(X23,Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_int_rat(H,X2)),aa_int_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8200_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_bool,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,zero_zero_rat),zero_zero_rat))
     => ( ! [X15: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,X15),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,plus_plus_rat(X15,Y1)),plus_plus_rat(X23,Y23))) )
       => ( finite3207457112153483333omplex(S2)
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,aa_complex_rat(H,X2)),aa_complex_rat(G,X2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R3,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8201_sum_Orelated,axiom,
    ! [R3: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S2))
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_int_nat(H,X2)),aa_int_nat(G,X2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8202_sum_Orelated,axiom,
    ! [R3: fun_nat_fun_nat_bool,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,zero_zero_nat),zero_zero_nat))
     => ( ! [X15: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,X15),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( finite3207457112153483333omplex(S2)
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,aa_complex_nat(H,X2)),aa_complex_nat(G,X2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R3,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8203_sum_Orelated,axiom,
    ! [R3: fun_int_fun_int_bool,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(R3,zero_zero_int),zero_zero_int))
     => ( ! [X15: int,Y1: int,X23: int,Y23: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(R3,X15),X23))
              & pp(aa_int_bool(aa_int_fun_int_bool(R3,Y1),Y23)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R3,aa_int_int(plus_plus_int(X15),Y1)),aa_int_int(plus_plus_int(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S2))
         => ( ! [X2: nat] :
                ( pp(aa_set_nat_bool(member_nat(X2),S2))
               => pp(aa_int_bool(aa_int_fun_int_bool(R3,aa_nat_int(H,X2)),aa_nat_int(G,X2))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(R3,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2))) ) ) ) ) ).

% sum.related
tff(fact_8204_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,A2)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_VEBT_VEBT_real(F,X2)),aa_VEBT_VEBT_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8205_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( member_complex(X2,A2)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8206_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8207_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8208_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,A2)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_VEBT_VEBT_rat(F,X2)),aa_VEBT_VEBT_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8209_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( member_complex(X2,A2)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8210_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8211_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8212_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8213_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,A2)
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_VEBT_VEBT_nat(F,X2)),aa_VEBT_VEBT_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups771621172384141258BT_nat(F,A2)),groups771621172384141258BT_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_8214_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = groups8097168146408367636l_real(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8215_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( member_VEBT_VEBT(X,A2)
         => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = groups2240296850493347238T_real(G,A2) ) )
        & ( ~ member_VEBT_VEBT(X,A2)
         => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8216_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = groups8778361861064173332t_real(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8217_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ( member_complex(X,A2)
         => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = groups5808333547571424918x_real(G,A2) ) )
        & ( ~ member_complex(X,A2)
         => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8218_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = groups1300246762558778688al_rat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = plus_plus_rat(aa_real_rat(G,X),groups1300246762558778688al_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8219_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( ( member_VEBT_VEBT(X,A2)
         => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = groups136491112297645522BT_rat(G,A2) ) )
        & ( ~ member_VEBT_VEBT(X,A2)
         => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8220_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = groups2906978787729119204at_rat(G,A2) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
         => ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = plus_plus_rat(aa_nat_rat(G,X),groups2906978787729119204at_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8221_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( pp(aa_set_int_bool(member_int(X),A2))
         => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = groups3906332499630173760nt_rat(G,A2) ) )
        & ( ~ pp(aa_set_int_bool(member_int(X),A2))
         => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8222_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( member_complex(X,A2)
         => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = groups5058264527183730370ex_rat(G,A2) ) )
        & ( ~ member_complex(X,A2)
         => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8223_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: fun_real_nat] :
      ( pp(aa_set_real_bool(finite_finite_real,A2))
     => ( ( pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = groups1935376822645274424al_nat(G,A2) ) )
        & ( ~ pp(aa_set_real_bool(member_real(X),A2))
         => ( groups1935376822645274424al_nat(G,insert_real(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,X)),groups1935376822645274424al_nat(G,A2)) ) ) ) ) ).

% sum.insert_if
tff(fact_8224_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,I: fun_real_real,J2: fun_real_real,T5: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_real_real(I,aa_real_real(J2,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J2,A4)),minus_minus_set_real(T5,T4))) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                 => ( aa_real_real(J2,aa_real_real(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B4)),minus_minus_set_real(S2,S4))) )
               => ( ! [A4: real] :
                      ( pp(aa_set_real_bool(member_real(A4),S4))
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( pp(aa_set_real_bool(member_real(B4),T4))
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_real_complex(H,aa_real_real(J2,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8225_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_VEBT_VEBT,S2: set_real,I: fun_VEBT_VEBT_real,J2: fun_real_VEBT_VEBT,T5: set_VEBT_VEBT,G: fun_real_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( finite5795047828879050333T_VEBT(T4)
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J2,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
               => member_VEBT_VEBT(aa_real_VEBT_VEBT(J2,A4),minus_5127226145743854075T_VEBT(T5,T4)) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(T5,T4))
                 => ( aa_real_VEBT_VEBT(J2,aa_VEBT_VEBT_real(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(T5,T4))
                   => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(I,B4)),minus_minus_set_real(S2,S4))) )
               => ( ! [A4: real] :
                      ( pp(aa_set_real_bool(member_real(A4),S4))
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(B4,T4)
                       => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_real_VEBT_VEBT(J2,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8226_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_real,S2: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J2: fun_VEBT_VEBT_real,T5: set_real,G: fun_VE5569070535936029429omplex,H: fun_real_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A4: vEBT_VEBT] :
              ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(S2,S4))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J2,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(S2,S4))
               => pp(aa_set_real_bool(member_real(aa_VEBT_VEBT_real(J2,A4)),minus_minus_set_real(T5,T4))) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                 => ( aa_VEBT_VEBT_real(J2,aa_real_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                   => member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B4),minus_5127226145743854075T_VEBT(S2,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(A4,S4)
                     => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( pp(aa_set_real_bool(member_real(B4),T4))
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(A4,S2)
                         => ( aa_real_complex(H,aa_VEBT_VEBT_real(J2,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups1794756597179926696omplex(G,S2) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8227_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J2: fun_VE3165209603783583043T_VEBT,T5: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( finite5795047828879050333T_VEBT(T4)
       => ( ! [A4: vEBT_VEBT] :
              ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(S2,S4))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J2,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(S2,S4))
               => member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J2,A4),minus_5127226145743854075T_VEBT(T5,T4)) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(T5,T4))
                 => ( aa_VEB5832953974749077948T_VEBT(J2,aa_VEB5832953974749077948T_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(T5,T4))
                   => member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B4),minus_5127226145743854075T_VEBT(S2,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(A4,S4)
                     => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(B4,T4)
                       => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(A4,S2)
                         => ( aa_VEBT_VEBT_complex(H,aa_VEB5832953974749077948T_VEBT(J2,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups1794756597179926696omplex(G,S2) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8228_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,I: fun_int_real,J2: fun_real_int,T5: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_int_real(I,aa_real_int(J2,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_real_int(J2,A4)),minus_minus_set_int(T5,T4))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T5,T4)))
                 => ( aa_real_int(J2,aa_int_real(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T5,T4)))
                   => pp(aa_set_real_bool(member_real(aa_int_real(I,B4)),minus_minus_set_real(S2,S4))) )
               => ( ! [A4: real] :
                      ( pp(aa_set_real_bool(member_real(A4),S4))
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T4))
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_int_complex(H,aa_real_int(J2,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups3049146728041665814omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8229_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T4: set_int,S2: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J2: fun_VEBT_VEBT_int,T5: set_int,G: fun_VE5569070535936029429omplex,H: fun_int_complex] :
      ( finite5795047828879050333T_VEBT(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A4: vEBT_VEBT] :
              ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(S2,S4))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J2,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(S2,S4))
               => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(J2,A4)),minus_minus_set_int(T5,T4))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T5,T4)))
                 => ( aa_VEBT_VEBT_int(J2,aa_int_VEBT_VEBT(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T5,T4)))
                   => member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B4),minus_5127226145743854075T_VEBT(S2,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(A4,S4)
                     => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T4))
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(A4,S2)
                         => ( aa_int_complex(H,aa_VEBT_VEBT_int(J2,A4)) = aa_VEBT_VEBT_complex(G,A4) ) )
                     => ( groups1794756597179926696omplex(G,S2) = groups3049146728041665814omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8230_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,I: fun_real_int,J2: fun_int_real,T5: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_real_int(I,aa_int_real(J2,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_int_real(J2,A4)),minus_minus_set_real(T5,T4))) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                 => ( aa_int_real(J2,aa_real_int(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B4)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( pp(aa_set_real_bool(member_real(B4),T4))
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_real_complex(H,aa_int_real(J2,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8231_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_VEBT_VEBT,S2: set_int,I: fun_VEBT_VEBT_int,J2: fun_int_VEBT_VEBT,T5: set_VEBT_VEBT,G: fun_int_complex,H: fun_VE5569070535936029429omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( finite5795047828879050333T_VEBT(T4)
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_VEBT_VEBT_int(I,aa_int_VEBT_VEBT(J2,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
               => member_VEBT_VEBT(aa_int_VEBT_VEBT(J2,A4),minus_5127226145743854075T_VEBT(T5,T4)) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(T5,T4))
                 => ( aa_int_VEBT_VEBT(J2,aa_VEBT_VEBT_int(I,B4)) = B4 ) )
             => ( ! [B4: vEBT_VEBT] :
                    ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(T5,T4))
                   => pp(aa_set_int_bool(member_int(aa_VEBT_VEBT_int(I,B4)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(B4,T4)
                       => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_VEBT_VEBT_complex(H,aa_int_VEBT_VEBT(J2,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8232_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S2: set_int,I: fun_int_int,J2: fun_int_int,T5: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T4))
       => ( ! [A4: int] :
              ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
             => ( aa_int_int(I,aa_int_int(J2,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J2,A4)),minus_minus_set_int(T5,T4))) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T5,T4)))
                 => ( aa_int_int(J2,aa_int_int(I,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T5,T4)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B4)),minus_minus_set_int(S2,S4))) )
               => ( ! [A4: int] :
                      ( pp(aa_set_int_bool(member_int(A4),S4))
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( pp(aa_set_int_bool(member_int(B4),T4))
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( pp(aa_set_int_bool(member_int(A4),S2))
                         => ( aa_int_complex(H,aa_int_int(J2,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8233_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,I: fun_real_real,J2: fun_real_real,T5: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,S4))
     => ( pp(aa_set_real_bool(finite_finite_real,T4))
       => ( ! [A4: real] :
              ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
             => ( aa_real_real(I,aa_real_real(J2,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
               => pp(aa_set_real_bool(member_real(aa_real_real(J2,A4)),minus_minus_set_real(T5,T4))) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                 => ( aa_real_real(J2,aa_real_real(I,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T5,T4)))
                   => pp(aa_set_real_bool(member_real(aa_real_real(I,B4)),minus_minus_set_real(S2,S4))) )
               => ( ! [A4: real] :
                      ( pp(aa_set_real_bool(member_real(A4),S4))
                     => ( aa_real_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( pp(aa_set_real_bool(member_real(B4),T4))
                       => ( aa_real_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( pp(aa_set_real_bool(member_real(A4),S2))
                         => ( aa_real_real(H,aa_real_real(J2,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8234_powr__non__neg,axiom,
    ! [A: real,X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real)) ).

% powr_non_neg
tff(fact_8235_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8236_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_mono2
tff(fact_8237_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ).

% powr_ge_pzero
tff(fact_8238_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel
tff(fact_8239_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))) ) ) ).

% powr_less_mono
tff(fact_8240_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8241_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( member_VEBT_VEBT(I3,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
       => ( ( groups136491112297645522BT_rat(F,S) = zero_zero_rat )
         => ( member_VEBT_VEBT(I,S)
           => ( aa_VEBT_VEBT_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8242_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
       => ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8243_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8244_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( finite3207457112153483333omplex(S)
     => ( ! [I3: complex] :
            ( member_complex(I3,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
         => ( member_complex(I,S)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8245_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_re6978948838062957830nteger,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,I3))) )
       => ( ( groups7713935264441627589nteger(F,S) = zero_z3403309356797280102nteger )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => ( aa_real_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8246_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( member_VEBT_VEBT(I3,S)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,I3))) )
       => ( ( groups5748017345553531991nteger(F,S) = zero_z3403309356797280102nteger )
         => ( member_VEBT_VEBT(I,S)
           => ( aa_VEB5935388676523804373nteger(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8247_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_Code_integer,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))) )
       => ( ( groups7501900531339628137nteger(F,S) = zero_z3403309356797280102nteger )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => ( aa_nat_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8248_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_Code_integer,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))) )
       => ( ( groups7873554091576472773nteger(F,S) = zero_z3403309356797280102nteger )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_Code_integer(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8249_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_co8153928105989009288nteger,I: complex] :
      ( finite3207457112153483333omplex(S)
     => ( ! [I3: complex] :
            ( member_complex(I3,S)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,I3))) )
       => ( ( groups6621422865394947399nteger(F,S) = zero_z3403309356797280102nteger )
         => ( member_complex(I,S)
           => ( aa_com4217951221296830281nteger(F,I) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8250_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
       => ( ( groups1300246762558778688al_rat(F,S) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8251_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,B2: rat,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( member_VEBT_VEBT(I3,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
       => ( ( groups136491112297645522BT_rat(F,S) = B2 )
         => ( member_VEBT_VEBT(I,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8252_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
       => ( ( groups2906978787729119204at_rat(F,S) = B2 )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8253_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8254_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( finite3207457112153483333omplex(S)
     => ( ! [I3: complex] :
            ( member_complex(I3,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = B2 )
         => ( member_complex(I,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8255_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_re6978948838062957830nteger,B2: code_integer,I: real] :
      ( pp(aa_set_real_bool(finite_finite_real,S))
     => ( ! [I3: real] :
            ( pp(aa_set_real_bool(member_real(I3),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,I3))) )
       => ( ( groups7713935264441627589nteger(F,S) = B2 )
         => ( pp(aa_set_real_bool(member_real(I),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_real_Code_integer(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8256_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger,B2: code_integer,I: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(S)
     => ( ! [I3: vEBT_VEBT] :
            ( member_VEBT_VEBT(I3,S)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,I3))) )
       => ( ( groups5748017345553531991nteger(F,S) = B2 )
         => ( member_VEBT_VEBT(I,S)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_VEB5935388676523804373nteger(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8257_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_Code_integer,B2: code_integer,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I3: nat] :
            ( pp(aa_set_nat_bool(member_nat(I3),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))) )
       => ( ( groups7501900531339628137nteger(F,S) = B2 )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8258_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_Code_integer,B2: code_integer,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I3: int] :
            ( pp(aa_set_int_bool(member_int(I3),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))) )
       => ( ( groups7873554091576472773nteger(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8259_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_co8153928105989009288nteger,B2: code_integer,I: complex] :
      ( finite3207457112153483333omplex(S)
     => ( ! [I3: complex] :
            ( member_complex(I3,S)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,I3))) )
       => ( ( groups6621422865394947399nteger(F,S) = B2 )
         => ( member_complex(I,S)
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8260_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ko(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8261_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kp(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8262_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kq(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8263_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kr(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8264_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_ks(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8265_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kt(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8266_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_ku(G)))) = groups5693394587270226106ex_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8267_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kv(G)))) = groups5690904116761175830ex_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8268_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7873554091576472773nteger(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kw(G)))) = groups7873554091576472773nteger(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8269_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(A2)
     => ( groups6621422865394947399nteger(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kx(G)))) = groups6621422865394947399nteger(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8270_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_8271_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_fg(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_8272_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8273_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_fg(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8274_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kz(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.shift_bounds_nat_ivl
tff(fact_8275_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_em(G),K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ).

% sum.shift_bounds_nat_ivl
tff(fact_8276_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kz(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8277_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_em(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8278_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8279_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( member_VEBT_VEBT(I,I5)
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( member_VEBT_VEBT(I3,I5)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8280_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8281_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8282_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I5)
     => ( member_complex(I,I5)
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
         => ( ! [I3: complex] :
                ( member_complex(I3,I5)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8283_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( pp(aa_set_real_bool(member_real(I),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,I)))
         => ( ! [I3: real] :
                ( pp(aa_set_real_bool(member_real(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7713935264441627589nteger(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8284_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( member_VEBT_VEBT(I,I5)
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,I)))
         => ( ! [I3: vEBT_VEBT] :
                ( member_VEBT_VEBT(I3,I5)
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups5748017345553531991nteger(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8285_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I)))
         => ( ! [I3: nat] :
                ( pp(aa_set_nat_bool(member_nat(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7501900531339628137nteger(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8286_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I)))
         => ( ! [I3: int] :
                ( pp(aa_set_int_bool(member_int(I3),I5))
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8287_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(I5)
     => ( member_complex(I,I5)
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,I)))
         => ( ! [I3: complex] :
                ( member_complex(I3,I5)
               => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,I3))) )
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups6621422865394947399nteger(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_8288_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( member_VEBT_VEBT(I3,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8289_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( member_complex(I3,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8290_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8291_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8292_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( member_VEBT_VEBT(I3,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8293_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( member_complex(I3,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8294_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8295_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8296_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,I5))
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8297_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( member_VEBT_VEBT(I3,I5)
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,I3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups771621172384141258BT_nat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_8298_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T5))
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(T5,S2)))
             => ( aa_real_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),S2))
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,T5) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8299_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(T5)
     => ( ord_le4337996190870823476T_VEBT(S2,T5)
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(T5,S2))
             => ( aa_VEBT_VEBT_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,S2)
               => ( aa_VEBT_VEBT_complex(G,X2) = aa_VEBT_VEBT_complex(H,X2) ) )
           => ( groups1794756597179926696omplex(G,T5) = groups1794756597179926696omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8300_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => ( aa_int_complex(G,X2) = aa_int_complex(H,X2) ) )
           => ( groups3049146728041665814omplex(G,T5) = groups3049146728041665814omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8301_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T5))
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(T5,S2)))
             => ( aa_real_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),S2))
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,T5) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8302_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(T5)
     => ( ord_le4337996190870823476T_VEBT(S2,T5)
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(T5,S2))
             => ( aa_VEBT_VEBT_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,S2)
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,T5) = groups2240296850493347238T_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8303_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => ( aa_int_real(G,X2) = aa_int_real(H,X2) ) )
           => ( groups8778361861064173332t_real(G,T5) = groups8778361861064173332t_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8304_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,T5) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8305_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T5))
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(T5,S2)))
             => ( aa_real_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),S2))
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,T5) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8306_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(T5)
     => ( ord_le4337996190870823476T_VEBT(S2,T5)
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(T5,S2))
             => ( aa_VEBT_VEBT_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,S2)
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,T5) = groups136491112297645522BT_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8307_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => ( aa_int_rat(G,X2) = aa_int_rat(H,X2) ) )
           => ( groups3906332499630173760nt_rat(G,T5) = groups3906332499630173760nt_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8308_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T5))
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(T5,S2)))
             => ( aa_real_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),S2))
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8309_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(T5)
     => ( ord_le4337996190870823476T_VEBT(S2,T5)
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(T5,S2))
             => ( aa_VEBT_VEBT_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,S2)
               => ( aa_VEBT_VEBT_complex(G,X2) = aa_VEBT_VEBT_complex(H,X2) ) )
           => ( groups1794756597179926696omplex(G,S2) = groups1794756597179926696omplex(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8310_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => ( aa_int_complex(G,X2) = aa_int_complex(H,X2) ) )
           => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8311_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T5))
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(T5,S2)))
             => ( aa_real_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),S2))
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8312_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(T5)
     => ( ord_le4337996190870823476T_VEBT(S2,T5)
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(T5,S2))
             => ( aa_VEBT_VEBT_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,S2)
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,S2) = groups2240296850493347238T_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8313_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => ( aa_int_real(G,X2) = aa_int_real(H,X2) ) )
           => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8314_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8315_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,T5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T5))
       => ( ! [X2: real] :
              ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(T5,S2)))
             => ( aa_real_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),S2))
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8316_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(T5)
     => ( ord_le4337996190870823476T_VEBT(S2,T5)
       => ( ! [X2: vEBT_VEBT] :
              ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(T5,S2))
             => ( aa_VEBT_VEBT_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,S2)
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,S2) = groups136491112297645522BT_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8317_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: int] :
                ( pp(aa_set_int_bool(member_int(X2),S2))
               => ( aa_int_rat(G,X2) = aa_int_rat(H,X2) ) )
           => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8318_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T5) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8319_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T5) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8320_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T5) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8321_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T5) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8322_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T5) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8323_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T5) = groups4541462559716669496nt_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8324_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T5) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8325_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T5) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8326_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_Code_integer(G,X2) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,T5) = groups7873554091576472773nteger(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8327_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_com4217951221296830281nteger(G,X2) = zero_z3403309356797280102nteger ) )
         => ( groups6621422865394947399nteger(G,T5) = groups6621422865394947399nteger(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8328_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8329_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8330_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8331_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8332_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8333_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8334_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8335_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8336_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,T5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T5))
       => ( ! [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(T5,S2)))
             => ( aa_int_Code_integer(G,X2) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(G,S2) = groups7873554091576472773nteger(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8337_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S2: set_complex,G: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(T5)
     => ( ord_le211207098394363844omplex(S2,T5)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T5,S2))
             => ( aa_com4217951221296830281nteger(G,X2) = zero_z3403309356797280102nteger ) )
         => ( groups6621422865394947399nteger(G,S2) = groups6621422865394947399nteger(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8338_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C5))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C5))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C5,A2)))
               => ( aa_real_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C5,B2)))
                 => ( aa_real_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C5) = groups5754745047067104278omplex(H,C5) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8339_sum_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(C5)
     => ( ord_le4337996190870823476T_VEBT(A2,C5)
       => ( ord_le4337996190870823476T_VEBT(B2,C5)
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(C5,B2))
                 => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,C5) = groups1794756597179926696omplex(H,C5) )
               => ( groups1794756597179926696omplex(G,A2) = groups1794756597179926696omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8340_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,C5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C5))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C5))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C5,A2)))
               => ( aa_int_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C5,B2)))
                 => ( aa_int_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,C5) = groups3049146728041665814omplex(H,C5) )
               => ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8341_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C5))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C5))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C5,A2)))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C5,B2)))
                 => ( aa_real_real(H,B4) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C5) = groups8097168146408367636l_real(H,C5) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8342_sum_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(C5)
     => ( ord_le4337996190870823476T_VEBT(A2,C5)
       => ( ord_le4337996190870823476T_VEBT(B2,C5)
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(C5,B2))
                 => ( aa_VEBT_VEBT_real(H,B4) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C5) = groups2240296850493347238T_real(H,C5) )
               => ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8343_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C5))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C5))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C5,A2)))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C5,B2)))
                 => ( aa_int_real(H,B4) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C5) = groups8778361861064173332t_real(H,C5) )
               => ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8344_sum_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C5)
     => ( ord_le211207098394363844omplex(A2,C5)
       => ( ord_le211207098394363844omplex(B2,C5)
         => ( ! [A4: complex] :
                ( member_complex(A4,minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( member_complex(B4,minus_811609699411566653omplex(C5,B2))
                 => ( aa_complex_real(H,B4) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C5) = groups5808333547571424918x_real(H,C5) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8345_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C5))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C5))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C5,A2)))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C5,B2)))
                 => ( aa_real_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C5) = groups1300246762558778688al_rat(H,C5) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8346_sum_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(C5)
     => ( ord_le4337996190870823476T_VEBT(A2,C5)
       => ( ord_le4337996190870823476T_VEBT(B2,C5)
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(C5,B2))
                 => ( aa_VEBT_VEBT_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C5) = groups136491112297645522BT_rat(H,C5) )
               => ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8347_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C5))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C5))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C5,A2)))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C5,B2)))
                 => ( aa_int_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C5) = groups3906332499630173760nt_rat(H,C5) )
               => ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8348_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( pp(aa_set_real_bool(finite_finite_real,C5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C5))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C5))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C5,A2)))
               => ( aa_real_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C5,B2)))
                 => ( aa_real_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C5) = groups5754745047067104278omplex(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8349_sum_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VE5569070535936029429omplex,H: fun_VE5569070535936029429omplex] :
      ( finite5795047828879050333T_VEBT(C5)
     => ( ord_le4337996190870823476T_VEBT(A2,C5)
       => ( ord_le4337996190870823476T_VEBT(B2,C5)
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(C5,B2))
                 => ( aa_VEBT_VEBT_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups1794756597179926696omplex(G,A2) = groups1794756597179926696omplex(H,B2) )
              <=> ( groups1794756597179926696omplex(G,C5) = groups1794756597179926696omplex(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8350_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,C5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C5))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C5))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C5,A2)))
               => ( aa_int_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C5,B2)))
                 => ( aa_int_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) )
              <=> ( groups3049146728041665814omplex(G,C5) = groups3049146728041665814omplex(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8351_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,C5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C5))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C5))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C5,A2)))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C5,B2)))
                 => ( aa_real_real(H,B4) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C5) = groups8097168146408367636l_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8352_sum_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(C5)
     => ( ord_le4337996190870823476T_VEBT(A2,C5)
       => ( ord_le4337996190870823476T_VEBT(B2,C5)
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(C5,B2))
                 => ( aa_VEBT_VEBT_real(H,B4) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A2) = groups2240296850493347238T_real(H,B2) )
              <=> ( groups2240296850493347238T_real(G,C5) = groups2240296850493347238T_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8353_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C5))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C5))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C5,A2)))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C5,B2)))
                 => ( aa_int_real(H,B4) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) )
              <=> ( groups8778361861064173332t_real(G,C5) = groups8778361861064173332t_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8354_sum_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C5)
     => ( ord_le211207098394363844omplex(A2,C5)
       => ( ord_le211207098394363844omplex(B2,C5)
         => ( ! [A4: complex] :
                ( member_complex(A4,minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( member_complex(B4,minus_811609699411566653omplex(C5,B2))
                 => ( aa_complex_real(H,B4) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C5) = groups5808333547571424918x_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8355_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,C5))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C5))
       => ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C5))
         => ( ! [A4: real] :
                ( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C5,A2)))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C5,B2)))
                 => ( aa_real_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C5) = groups1300246762558778688al_rat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8356_sum_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(C5)
     => ( ord_le4337996190870823476T_VEBT(A2,C5)
       => ( ord_le4337996190870823476T_VEBT(B2,C5)
         => ( ! [A4: vEBT_VEBT] :
                ( member_VEBT_VEBT(A4,minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(C5,B2))
                 => ( aa_VEBT_VEBT_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A2) = groups136491112297645522BT_rat(H,B2) )
              <=> ( groups136491112297645522BT_rat(G,C5) = groups136491112297645522BT_rat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8357_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C5))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C5))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C5))
         => ( ! [A4: int] :
                ( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C5,A2)))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C5,B2)))
                 => ( aa_int_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) )
              <=> ( groups3906332499630173760nt_rat(G,C5) = groups3906332499630173760nt_rat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8358_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8359_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( ord_le211207098394363844omplex(B2,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8360_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups3906332499630173760nt_rat(G,A2) = plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8361_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( ord_le211207098394363844omplex(B2,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5058264527183730370ex_rat(G,A2) = plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2)),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8362_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8363_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( ord_le211207098394363844omplex(B2,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8364_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( ord_le211207098394363844omplex(B2,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8365_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups2906978787729119204at_rat(G,A2) = plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2)),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8366_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8367_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),minus_minus_set_nat(A2,B2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8368_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),D))
               => ( aa_nat_nat(G,X2) = aa_nat_nat(H,X2) ) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = aa_set_nat_nat(groups3542108847815614940at_nat(H),aa_nat_set_nat(set_or4665077453230672383an_nat(C),D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_8369_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_real,H: fun_nat_real] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),D))
               => ( aa_nat_real(G,X2) = aa_nat_real(H,X2) ) ) )
         => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = groups6591440286371151544t_real(H,aa_nat_set_nat(set_or4665077453230672383an_nat(C),D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_8370_sum_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D: int,G: fun_int_int,H: fun_int_int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),X2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),D))
               => ( aa_int_int(G,X2) = aa_int_int(H,X2) ) ) )
         => ( groups4538972089207619220nt_int(G,set_or4662586982721622107an_int(A,B)) = groups4538972089207619220nt_int(H,set_or4662586982721622107an_int(C,D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_8371_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_mono2'
tff(fact_8372_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_less_mono2
tff(fact_8373_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_8374_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ) ) ).

% gr_one_powr
tff(fact_8375_powr__le1,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_8376_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B))) ) ) ) ) ).

% powr_mono_both
tff(fact_8377_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8378_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_8379_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_8380_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8381_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8382_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8383_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P3: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3))) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8384_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P3: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)),groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8385_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P3: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( aa_int_int(minus_minus_int(groups3539618377306564664at_int(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3))),groups3539618377306564664at_int(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8386_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P3: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),P3))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(N),P3)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8387_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_8388_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),B) ) ) ).

% log_base_powr
tff(fact_8389_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_powr
tff(fact_8390_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_powr
tff(fact_8391_powr__add,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)) ).

% powr_add
tff(fact_8392_powr__diff,axiom,
    ! [W: real,Z1: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z2)) ).

% powr_diff
tff(fact_8393_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_la(X),M),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),I5)) ).

% sum_power_add
tff(fact_8394_sum__power__add,axiom,
    ! [X: code_integer,M: nat,I5: set_nat] : groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_lb(X),M),I5) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),I5)) ).

% sum_power_add
tff(fact_8395_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_lc(X),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_8396_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ld(X),M),I5) = aa_int_int(times_times_int(aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_8397_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_le(X),M),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),I5)) ).

% sum_power_add
tff(fact_8398_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B4: real] :
              ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8399_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [B4: vEBT_VEBT] :
              ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(B2,A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8400_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B4: int] :
              ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8401_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B2)
     => ( ord_le211207098394363844omplex(A2,B2)
       => ( ! [B4: complex] :
              ( member_complex(B4,minus_811609699411566653omplex(B2,A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8402_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B4: real] :
              ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,B4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7713935264441627589nteger(F,A2)),groups7713935264441627589nteger(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8403_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [B4: vEBT_VEBT] :
              ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(B2,A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,B4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups5748017345553531991nteger(F,A2)),groups5748017345553531991nteger(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8404_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B4: int] :
              ( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,B4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A2)),groups7873554091576472773nteger(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8405_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(B2)
     => ( ord_le211207098394363844omplex(A2,B2)
       => ( ! [B4: complex] :
              ( member_complex(B4,minus_811609699411566653omplex(B2,A2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,B4))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups6621422865394947399nteger(F,A2)),groups6621422865394947399nteger(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8406_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( ! [B4: real] :
              ( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8407_sum__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( ! [B4: vEBT_VEBT] :
              ( member_VEBT_VEBT(B4,minus_5127226145743854075T_VEBT(B2,A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,B4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_8408_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lf(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8409_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lg(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8410_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( member_VEBT_VEBT(X,A2)
       => ( groups2240296850493347238T_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_8411_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( member_complex(X,A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8412_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( member_VEBT_VEBT(X,A2)
       => ( groups136491112297645522BT_rat(G,A2) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_8413_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( member_complex(X,A2)
       => ( groups5058264527183730370ex_rat(G,A2) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8414_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( member_VEBT_VEBT(X,A2)
       => ( groups771621172384141258BT_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_8415_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( member_complex(X,A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8416_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_int] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( member_VEBT_VEBT(X,A2)
       => ( groups769130701875090982BT_int(G,A2) = aa_int_int(plus_plus_int(aa_VEBT_VEBT_int(G,X)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_8417_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A2)
     => ( member_complex(X,A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(plus_plus_int(aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8418_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8419_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => ( groups3906332499630173760nt_rat(G,A2) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8420_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,X: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_8421_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,X: complex] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8422_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,X: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_8423_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,X: complex] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8424_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,X: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_8425_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,X: complex] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8426_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,X: vEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ( groups769130701875090982BT_int(G,insert_VEBT_VEBT(X,A2)) = aa_int_int(plus_plus_int(aa_VEBT_VEBT_int(G,X)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_8427_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,X: complex] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(plus_plus_int(aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8428_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_real,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8429_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_rat,X: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8430_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_complex(F,N2) = zero_zero_complex ) )
       => ( suminf_complex(F) = groups2073611262835488442omplex(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8431_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_int(F,N2) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8432_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_nat(F,N2) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N4) ) ) ) ).

% suminf_finite
tff(fact_8433_suminf__finite,axiom,
    ! [N4: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N4))
     => ( ! [N2: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N2),N4))
           => ( aa_nat_real(F,N2) = zero_zero_real ) )
       => ( suminf_real(F) = groups6591440286371151544t_real(F,N4) ) ) ) ).

% suminf_finite
tff(fact_8434_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_lh(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B,A)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_lh(A),B),C),S2) = groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8435_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( finite3207457112153483333omplex(S2)
     => ( ( member_complex(A,S2)
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_li(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ member_complex(A,S2)
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_li(A),B),C),S2) = groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8436_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_lj(A),B),C),S2) = plus_plus_rat(aa_VEBT_VEBT_rat(B,A),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_lj(A),B),C),S2) = groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8437_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( finite3207457112153483333omplex(S2)
     => ( ( member_complex(A,S2)
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_lk(A),B),C),S2) = plus_plus_rat(aa_complex_rat(B,A),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ member_complex(A,S2)
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_lk(A),B),C),S2) = groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8438_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_ll(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,A)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_ll(A),B),C),S2) = groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8439_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( finite3207457112153483333omplex(S2)
     => ( ( member_complex(A,S2)
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_lm(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ member_complex(A,S2)
         => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_lm(A),B),C),S2) = groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8440_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
      ( finite5795047828879050333T_VEBT(S2)
     => ( ( member_VEBT_VEBT(A,S2)
         => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_ln(A),B),C),S2) = aa_int_int(plus_plus_int(aa_VEBT_VEBT_int(B,A)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
        & ( ~ member_VEBT_VEBT(A,S2)
         => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_ln(A),B),C),S2) = groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8441_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
      ( finite3207457112153483333omplex(S2)
     => ( ( member_complex(A,S2)
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lo(A),B),C),S2) = aa_int_int(plus_plus_int(aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
        & ( ~ member_complex(A,S2)
         => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lo(A),B),C),S2) = groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8442_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_real,C: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lp(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(B,A)),groups8778361861064173332t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lp(A),B),C),S2) = groups8778361861064173332t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8443_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S2))
     => ( ( pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_lq(A),B),C),S2) = plus_plus_rat(aa_int_rat(B,A),groups3906332499630173760nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S2))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_lq(A),B),C),S2) = groups3906332499630173760nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).

% sum.delta_remove
tff(fact_8444_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8445_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8446_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8447_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_Code_integer,K: nat] :
      ( ( aa_nat_Code_integer(F,zero_zero_nat) = zero_z3403309356797280102nteger )
     => ( groups7501900531339628137nteger(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups7501900531339628137nteger(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8448_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8449_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_8450_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8451_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8452_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8453_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_Code_integer,K: nat] :
      ( ( aa_nat_Code_integer(F,zero_zero_nat) = zero_z3403309356797280102nteger )
     => ( groups7501900531339628137nteger(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups7501900531339628137nteger(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8454_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8455_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8456_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)),aa_nat_rat(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8457_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_int(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8458_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_nat(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8459_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_nat_real(G,N)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8460_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8461_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_int_int(plus_plus_int(aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8462_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8463_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8464_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8465_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8466_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8467_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8468_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)),aa_nat_rat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8469_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_int(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8470_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_nat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8471_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_real(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8472_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N) ) ) ).

% powr_realpow
tff(fact_8473_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8474_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_nat_int(G,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8475_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8476_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8477_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8478_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(plus_plus_int(aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8479_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8480_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8481_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_less_iff
tff(fact_8482_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_8483_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_less_iff
tff(fact_8484_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_8485_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = plus_plus_rat(aa_nat_rat(G,N),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_8486_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(plus_plus_int(aa_nat_int(G,N)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_8487_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,N)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_8488_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,N)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))) ) ) ).

% sum.last_plus
tff(fact_8489_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B)))
           => ( ! [X2: real] :
                  ( pp(aa_set_real_bool(member_real(X2),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8490_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( member_VEBT_VEBT(B,minus_5127226145743854075T_VEBT(B2,A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B)))
           => ( ! [X2: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X2,B2)
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups136491112297645522BT_rat(F,A2)),groups136491112297645522BT_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8491_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,B)))
           => ( ! [X2: int] :
                  ( pp(aa_set_int_bool(member_int(X2),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8492_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B2)
     => ( ord_le211207098394363844omplex(A2,B2)
       => ( member_complex(B,minus_811609699411566653omplex(B2,A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B)))
           => ( ! [X2: complex] :
                  ( member_complex(X2,B2)
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8493_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,B)))
           => ( ! [X2: real] :
                  ( pp(aa_set_real_bool(member_real(X2),B2))
                 => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,X2))) )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups7713935264441627589nteger(F,A2)),groups7713935264441627589nteger(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8494_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( member_VEBT_VEBT(B,minus_5127226145743854075T_VEBT(B2,A2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,B)))
           => ( ! [X2: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X2,B2)
                 => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,X2))) )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups5748017345553531991nteger(F,A2)),groups5748017345553531991nteger(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8495_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,B)))
           => ( ! [X2: int] :
                  ( pp(aa_set_int_bool(member_int(X2),B2))
                 => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X2))) )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups7873554091576472773nteger(F,A2)),groups7873554091576472773nteger(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8496_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(B2)
     => ( ord_le211207098394363844omplex(A2,B2)
       => ( member_complex(B,minus_811609699411566653omplex(B2,A2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,B)))
           => ( ! [X2: complex] :
                  ( member_complex(X2,B2)
                 => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,X2))) )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,groups6621422865394947399nteger(F,A2)),groups6621422865394947399nteger(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8497_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( pp(aa_set_real_bool(finite_finite_real,B2))
     => ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
       => ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B)))
           => ( ! [X2: real] :
                  ( pp(aa_set_real_bool(member_real(X2),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8498_sum__strict__mono2,axiom,
    ! [B2: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( finite5795047828879050333T_VEBT(B2)
     => ( ord_le4337996190870823476T_VEBT(A2,B2)
       => ( member_VEBT_VEBT(B,minus_5127226145743854075T_VEBT(B2,A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,B)))
           => ( ! [X2: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X2,B2)
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X2))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2240296850493347238T_real(F,A2)),groups2240296850493347238T_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8499_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( member_VEBT_VEBT(I,A2)
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X2))) )
       => ( finite5795047828879050333T_VEBT(A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8500_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_complex_rat] :
      ( member_complex(I,A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2))) )
       => ( finite3207457112153483333omplex(A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8501_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( member_VEBT_VEBT(I,A2)
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(F,X2))) )
       => ( finite5795047828879050333T_VEBT(A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_VEB5935388676523804373nteger(F,I)),groups5748017345553531991nteger(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8502_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: fun_co8153928105989009288nteger] :
      ( member_complex(I,A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex)))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(F,X2))) )
       => ( finite3207457112153483333omplex(A2)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F,I)),groups6621422865394947399nteger(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8503_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8504_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: fun_int_Code_integer] :
      ( pp(aa_set_int_bool(member_int(I),A2))
     => ( ! [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X2))) )
       => ( pp(aa_set_int_bool(finite_finite_int,A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I)),groups7873554091576472773nteger(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8505_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_real_rat] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8506_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: fun_re6978948838062957830nteger] :
      ( pp(aa_set_real_bool(member_real(I),A2))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(F,X2))) )
       => ( pp(aa_set_real_bool(finite_finite_real,A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_real_Code_integer(F,I)),groups7713935264441627589nteger(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8507_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(A2,insert_nat(I,bot_bot_set_nat))))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8508_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: fun_nat_Code_integer] :
      ( pp(aa_set_nat_bool(member_nat(I),A2))
     => ( ! [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),minus_minus_set_nat(A2,insert_nat(I,bot_bot_set_nat))))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X2))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I)),groups7501900531339628137nteger(F,A2))) ) ) ) ).

% member_le_sum
tff(fact_8509_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(aTP_Lamm_lr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8510_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_ls(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8511_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8512_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(aTP_Lamm_fg(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8513_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_lt(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,N),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_8514_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_lu(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_int_int(minus_minus_int(aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_8515_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(aTP_Lamm_lv(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_8516_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_lt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N)),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8517_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups3539618377306564664at_int(aTP_Lamm_lu(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8518_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
     => ( groups6591440286371151544t_real(aTP_Lamm_lv(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_8519_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lw(G),N),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ).

% sum.atLeastLessThan_rev
tff(fact_8520_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lx(G),N),M),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) ).

% sum.atLeastLessThan_rev
tff(fact_8521_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I3))) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ly(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8522_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I3))) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_lz(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8523_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VE9219535225822561060nteger,A: fun_VE9219535225822561060nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: vEBT_VEBT] :
          ( member_VEBT_VEBT(I3,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_VEB5935388676523804373nteger(X,I3))) )
     => ( ( groups5748017345553531991nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: vEBT_VEBT] :
              ( member_VEBT_VEBT(I3,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_VEB5935388676523804373nteger(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups5748017345553531991nteger(aa_fun2400605388621188028nteger(aTP_Lamm_ma(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8524_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I3))) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_mb(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8525_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I3))) )
     => ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( member_complex(I3,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I3),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_mc(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8526_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( pp(aa_set_nat_bool(member_nat(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I3))) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I3: nat] :
              ( pp(aa_set_nat_bool(member_nat(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(A,I3),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_md(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8527_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I3: real] :
          ( pp(aa_set_real_bool(member_real(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I3))) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I3: real] :
              ( pp(aa_set_real_bool(member_real(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_real_rat(A,I3),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_me(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8528_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X: fun_VEBT_VEBT_rat,A: fun_VEBT_VEBT_rat,B: rat,Delta: rat] :
      ( ! [I3: vEBT_VEBT] :
          ( member_VEBT_VEBT(I3,I5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(X,I3))) )
     => ( ( groups136491112297645522BT_rat(X,I5) = one_one_rat )
       => ( ! [I3: vEBT_VEBT] :
              ( member_VEBT_VEBT(I3,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_VEBT_VEBT_rat(A,I3),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_mf(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8529_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I3: int] :
          ( pp(aa_set_int_bool(member_int(I3),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(X,I3))) )
     => ( ( groups3906332499630173760nt_rat(X,I5) = one_one_rat )
       => ( ! [I3: int] :
              ( pp(aa_set_int_bool(member_int(I3),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(A,I3),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mg(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8530_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X,I3))) )
     => ( ( groups5058264527183730370ex_rat(X,I5) = one_one_rat )
       => ( ! [I3: complex] :
              ( member_complex(I3,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_complex_rat(A,I3),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_mh(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_8531_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mi(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk(A),N)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% sum.nested_swap
tff(fact_8532_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_real,N: nat] : groups6591440286371151544t_real(aTP_Lamm_ml(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mn(A),N),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).

% sum.nested_swap
tff(fact_8533_powr__minus__divide,axiom,
    ! [X: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)) ).

% powr_minus_divide
tff(fact_8534_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_8535_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_8536_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_le_iff
tff(fact_8537_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_8538_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_le_iff
tff(fact_8539_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_8540_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8541_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8542_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8543_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_8544_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_complex(G,N)) ) ) ) ).

% sum.head_if
tff(fact_8545_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)),aa_nat_rat(G,N)) ) ) ) ).

% sum.head_if
tff(fact_8546_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_int(G,N)) ) ) ) ).

% sum.head_if
tff(fact_8547_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_Code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups7501900531339628137nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups7501900531339628137nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = plus_p5714425477246183910nteger(groups7501900531339628137nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)),aa_nat_Code_integer(G,N)) ) ) ) ).

% sum.head_if
tff(fact_8548_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_nat(G,N)) ) ) ) ).

% sum.head_if
tff(fact_8549_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
       => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N))),aa_nat_real(G,N)) ) ) ) ).

% sum.head_if
tff(fact_8550_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I5: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,I5)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8551_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I5: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8552_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I5: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N2: nat] :
              ( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,I5)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_8553_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% set_encode_def
tff(fact_8554_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lf(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8555_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lg(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8556_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_8557_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_8558_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8559_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8560_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8561_summable__Cauchy,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [N10: nat] :
            ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N10),M5))
             => ! [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M5),N7)))),E3)) ) ) ) ).

% summable_Cauchy
tff(fact_8562_summable__Cauchy,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
    <=> ! [E3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E3))
         => ? [N10: nat] :
            ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N10),M5))
             => ! [N7: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M5),N7)))),E3)) ) ) ) ).

% summable_Cauchy
tff(fact_8563_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_mo(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F,M)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2073611262835488442omplex(aTP_Lamm_mo(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
tff(fact_8564_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_Code_integer] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups7501900531339628137nteger(aTP_Lamm_mp(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_8373710615458151222nteger(aa_nat_Code_integer(F,M),aa_nat_Code_integer(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups7501900531339628137nteger(aTP_Lamm_mp(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_z3403309356797280102nteger ) ) ) ).

% sum_natinterval_diff
tff(fact_8565_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_mq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups2906978787729119204at_rat(aTP_Lamm_mq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_8566_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_mr(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(minus_minus_int(aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups3539618377306564664at_int(aTP_Lamm_mr(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_8567_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups6591440286371151544t_real(aTP_Lamm_ms(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( groups6591440286371151544t_real(aTP_Lamm_ms(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_8568_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_mt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = minus_minus_rat(aa_nat_rat(F,N),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_8569_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups3539618377306564664at_int(aTP_Lamm_mu(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(minus_minus_int(aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_8570_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( groups6591440286371151544t_real(aTP_Lamm_mv(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_8571_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N9)))),E)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8572_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N9)))),E)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8573_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8574_mask__eq__sum__exp,axiom,
    ! [N: nat] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N),one_one_Code_integer) = groups7501900531339628137nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,N))) ).

% mask_eq_sum_exp
tff(fact_8575_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,N))) ).

% mask_eq_sum_exp
tff(fact_8576_mask__eq__sum__exp,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,N))) ).

% mask_eq_sum_exp
tff(fact_8577_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8578_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),M),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8579_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8580_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(times_times_int(aa_int_int(minus_minus_int(one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8581_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ) ) ).

% sum_gp_multiplied
tff(fact_8582_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_mw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8583_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_mx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8584_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_my(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8585_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) = groups6591440286371151544t_real(aTP_Lamm_mz(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).

% sum.in_pairs
tff(fact_8586_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_8587_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,N))) ).

% mask_eq_sum_exp_nat
tff(fact_8588_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum_nat
tff(fact_8589_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)),one_one_nat) ).

% sum_power2
tff(fact_8590_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Sum_Ico_nat
tff(fact_8591_double__gauss__sum,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)) ).

% double_gauss_sum
tff(fact_8592_double__gauss__sum,axiom,
    ! [N: nat] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum
tff(fact_8593_double__gauss__sum,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum
tff(fact_8594_double__gauss__sum,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum
tff(fact_8595_double__gauss__sum,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum
tff(fact_8596_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_na(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),A),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).

% double_arith_series
tff(fact_8597_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nb(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).

% double_arith_series
tff(fact_8598_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_nc(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).

% double_arith_series
tff(fact_8599_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nd(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).

% double_arith_series
tff(fact_8600_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ne(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).

% double_arith_series
tff(fact_8601_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nf(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% arith_series_nat
tff(fact_8602_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% Sum_Icc_nat
tff(fact_8603_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8604_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8605_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8606_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8607_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_8608_gauss__sum,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% gauss_sum
tff(fact_8609_gauss__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum
tff(fact_8610_arith__series,axiom,
    ! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nb(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% arith_series
tff(fact_8611_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nd(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% arith_series
tff(fact_8612_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),minus_minus_rat(one_one_rat,X)) ) ) ) ).

% sum_gp_offset
tff(fact_8613_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8614_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_8615_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% gauss_sum_from_Suc_0
tff(fact_8616_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% gauss_sum_from_Suc_0
tff(fact_8617_Chebyshev__sum__upper,axiom,
    ! [N: nat,A: fun_nat_rat,B: fun_nat_rat] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(A,I3)),aa_nat_rat(A,J))) ) )
     => ( ! [I3: nat,J: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(B,J)),aa_nat_rat(B,I3))) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ng(A),B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(A,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),groups2906978787729119204at_rat(B,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper
tff(fact_8618_Chebyshev__sum__upper,axiom,
    ! [N: nat,A: fun_nat_int,B: fun_nat_int] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(A,I3)),aa_nat_int(A,J))) ) )
     => ( ! [I3: nat,J: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(B,J)),aa_nat_int(B,I3))) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_nh(A),B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),aa_int_int(times_times_int(groups3539618377306564664at_int(A,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),groups3539618377306564664at_int(B,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper
tff(fact_8619_Chebyshev__sum__upper,axiom,
    ! [N: nat,A: fun_nat_real,B: fun_nat_real] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,I3)),aa_nat_real(A,J))) ) )
     => ( ! [I3: nat,J: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(B,J)),aa_nat_real(B,I3))) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ni(A),B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(A,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),groups6591440286371151544t_real(B,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper
tff(fact_8620_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J))) ) )
     => ( ! [I3: nat,J: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),J))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),N))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J)),aa_nat_nat(B,I3))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nj(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8621_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Z,H)),N),aa_nat_rat(power_power_rat(Z),N)),H),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nl(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8622_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8623_lemma__termdiff2,axiom,
    ! [H: real,Z: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_np(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8624_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
     => ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ).

% sin_tan
tff(fact_8625_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,X) = aa_real_real(sqrt,Y) )
    <=> ( X = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_8626_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( pp(aa_set_rat_bool(member_rat(I),set_ord_lessThan_rat(K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),K)) ) ).

% lessThan_iff
tff(fact_8627_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( pp(aa_set_num_bool(member_num(I),set_ord_lessThan_num(K)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),K)) ) ).

% lessThan_iff
tff(fact_8628_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).

% lessThan_iff
tff(fact_8629_lessThan__iff,axiom,
    ! [I: code_integer,K: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(member_Code_integer(I),set_or5754767410780653050nteger(K)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),K)) ) ).

% lessThan_iff
tff(fact_8630_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_8631_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( pp(aa_set_real_bool(member_real(I),set_or5984915006950818249n_real(K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_8632_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_8633_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_8634_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_8635_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% real_sqrt_le_iff
tff(fact_8636_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_8637_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_8638_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_8639_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_8640_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_8641_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_8642_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_8643_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_8644_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_8645_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).

% real_sqrt_le_1_iff
tff(fact_8646_lessThan__minus__lessThan,axiom,
    ! [N: real,M: real] : minus_minus_set_real(set_or5984915006950818249n_real(N),set_or5984915006950818249n_real(M)) = set_or66887138388493659n_real(M,N) ).

% lessThan_minus_lessThan
tff(fact_8647_lessThan__minus__lessThan,axiom,
    ! [N: nat,M: nat] : minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_nat_set_nat(set_or4665077453230672383an_nat(M),N) ).

% lessThan_minus_lessThan
tff(fact_8648_lessThan__minus__lessThan,axiom,
    ! [N: int,M: int] : minus_minus_set_int(set_ord_lessThan_int(N),set_ord_lessThan_int(M)) = set_or4662586982721622107an_int(M,N) ).

% lessThan_minus_lessThan
tff(fact_8649_lessThan__minus__lessThan,axiom,
    ! [N: code_integer,M: code_integer] : minus_2355218937544613996nteger(set_or5754767410780653050nteger(N),set_or5754767410780653050nteger(M)) = set_or8404916559141939852nteger(M,N) ).

% lessThan_minus_lessThan
tff(fact_8650_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8651_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_8652_real__sqrt__abs2,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs2
tff(fact_8653_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(bit0(bit0(one)))) = numeral_numeral_real(bit0(one)) ).

% real_sqrt_four
tff(fact_8654_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_nat_rat(G,N)) ).

% sum.lessThan_Suc
tff(fact_8655_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).

% sum.lessThan_Suc
tff(fact_8656_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).

% sum.lessThan_Suc
tff(fact_8657_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).

% sum.lessThan_Suc
tff(fact_8658_real__sqrt__abs,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs
tff(fact_8659_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_8660_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,bit0(one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_8661_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa4: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa4),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa4),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_8662_Complex__sum_H,axiom,
    ! [F: fun_nat_real,S: set_nat] : groups2073611262835488442omplex(aTP_Lamm_nq(F),S) = aa_real_complex(complex2(groups6591440286371151544t_real(F,S)),zero_zero_real) ).

% Complex_sum'
tff(fact_8663_Complex__sum_H,axiom,
    ! [F: fun_complex_real,S: set_complex] : groups7754918857620584856omplex(aTP_Lamm_nr(F),S) = aa_real_complex(complex2(groups5808333547571424918x_real(F,S)),zero_zero_real) ).

% Complex_sum'
tff(fact_8664_real__sqrt__minus,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,X)) ).

% real_sqrt_minus
tff(fact_8665_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_hx(F),A2) ).

% int_sum
tff(fact_8666_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_hy(F),A2) ).

% int_sum
tff(fact_8667_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_8668_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_8669_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_le_mono
tff(fact_8670_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_8671_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),K) ).

% real_sqrt_power
tff(fact_8672_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_ns,U)) ).

% lessThan_def
tff(fact_8673_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_nt,U)) ).

% lessThan_def
tff(fact_8674_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_nu,U)) ).

% lessThan_def
tff(fact_8675_lessThan__def,axiom,
    ! [U: code_integer] : set_or5754767410780653050nteger(U) = collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_nv,U)) ).

% lessThan_def
tff(fact_8676_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,U)) ).

% lessThan_def
tff(fact_8677_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_nw,U)) ).

% lessThan_def
tff(fact_8678_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_8679_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_8680_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_8681_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_8682_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_8683_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( pp(aa_set_rat_bool(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(N)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8684_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( pp(aa_set_num_bool(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(N)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8685_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8686_lessThan__strict__subset__iff,axiom,
    ! [M: code_integer,N: code_integer] :
      ( pp(aa_set1374477347934578310r_bool(ord_le1307284697595431911nteger(set_or5754767410780653050nteger(M)),set_or5754767410780653050nteger(N)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8687_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8688_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(N)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N)) ) ).

% lessThan_strict_subset_iff
tff(fact_8689_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8690_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_8691_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N) )
     => ? [X2: nat] :
          ( pp(aa_set_nat_bool(member_nat(X2),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2))) ) ) ).

% sum_SucD
tff(fact_8692_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & ( aa_int_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8693_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: complex] :
            ( member_complex(X5,A2)
            & ( aa_complex_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( member_complex(Xa3,A2)
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8694_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & ( aa_nat_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8695_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X5: int] :
            ( pp(aa_set_int_bool(member_int(X5),A2))
            & ( aa_int_nat(F,X5) = one_one_nat )
            & ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8696_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X5: complex] :
            ( member_complex(X5,A2)
            & ( aa_complex_nat(F,X5) = one_one_nat )
            & ! [Xa3: complex] :
                ( member_complex(Xa3,A2)
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8697_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X5: nat] :
            ( pp(aa_set_nat_bool(member_nat(X5),A2))
            & ( aa_nat_nat(F,X5) = one_one_nat )
            & ! [Xa3: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa3),A2))
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8698_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_8699_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_8700_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))) ).

% le_real_sqrt_sumsq
tff(fact_8701_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8702_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nx(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_8703_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ny(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.nat_diff_reindex
tff(fact_8704_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one)))) ).

% sqrt2_less_2
tff(fact_8705_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,zero_zero_nat),groups2906978787729119204at_rat(aTP_Lamm_lr(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8706_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(plus_plus_int(aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ls(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8707_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8708_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_fg(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum.lessThan_Suc_shift
tff(fact_8709_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_nz(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8710_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_oa(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(minus_minus_int(aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8711_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_ob(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8712_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_lt(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8713_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_lu(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(minus_minus_int(aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8714_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_lv(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8715_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,N: nat,R2: rat] : minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_oc(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8716_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,N: nat,R2: int] : aa_int_int(minus_minus_int(groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_od(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8717_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_complex,N: nat,R2: complex] : aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),R2)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_oe(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8718_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,N: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),R2)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_of(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sumr_diff_mult_const2
tff(fact_8719_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: int] :
      ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
     => ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8720_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
     => ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8721_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8722_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_8723_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = groups6591440286371151544t_real(aTP_Lamm_fg(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).

% sum.atLeast1_atMost_eq
tff(fact_8724_sum_Onat__group,axiom,
    ! [G: fun_nat_nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_og(G),K)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% sum.nat_group
tff(fact_8725_sum_Onat__group,axiom,
    ! [G: fun_nat_real,K: nat,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oh(G),K),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))) ).

% sum.nat_group
tff(fact_8726_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_oi,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_ec(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8727_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_8728_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,Y))) ) ).

% real_le_rsqrt
tff(fact_8729_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ).

% sqrt_le_D
tff(fact_8730_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8731_power__diff__1__eq,axiom,
    ! [X: code_integer,N: nat] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N),one_one_Code_integer) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(X,one_one_Code_integer)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8732_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),N),one_one_rat) = aa_rat_rat(times_times_rat(minus_minus_rat(X,one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8733_power__diff__1__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),one_one_int)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8734_power__diff__1__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_1_eq
tff(fact_8735_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8736_one__diff__power__eq,axiom,
    ! [X: code_integer,N: nat] : minus_8373710615458151222nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(power_8256067586552552935nteger(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8737_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8738_one__diff__power__eq,axiom,
    ! [X: int,N: nat] : aa_int_int(minus_minus_int(one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8739_one__diff__power__eq,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq
tff(fact_8740_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_8741_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N),one_one_rat),minus_minus_rat(X,one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8742_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8743_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_em(F),K))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_8744_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_em(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_8745_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))) = X )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_8746_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_8747_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(bit0(one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_8748_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_8749_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_8750_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_8751_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_8752_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_8753_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_8754_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( groups7754918857620584856omplex(aTP_Lamm_oi,collect_complex(aTP_Lamm_dv(N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8755_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% cos_45
tff(fact_8756_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% sin_45
tff(fact_8757_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_8758_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M3))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_8759_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_8760_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M3))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_8761_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oj(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ok(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8762_lemma__termdiff1,axiom,
    ! [Z: code_integer,H: code_integer,M: nat] : groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_ol(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_om(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8763_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_on(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_oo(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8764_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_op(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_oq(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8765_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_or(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_os(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_8766_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)),minus_minus_rat(one_one_rat,X)) ) ) ) ).

% sum_gp_strict
tff(fact_8767_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_8768_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
      & ( ( X != one_one_real )
       => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_8769_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ot(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8770_diff__power__eq__sum,axiom,
    ! [X: code_integer,N: nat,Y: code_integer] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_nat_nat(suc,N))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(X,Y)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_ou(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8771_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ov(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8772_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ow(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8773_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ox(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).

% diff_power_eq_sum
tff(fact_8774_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_oy(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8775_power__diff__sumr2,axiom,
    ! [X: code_integer,N: nat,Y: code_integer] : minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(X),N),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(X,Y)),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_oz(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8776_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),N),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_pa(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8777_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_pb(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8778_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_pc(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% power_diff_sumr2
tff(fact_8779_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_8780_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_8781_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8782_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% sqrt_even_pow2
tff(fact_8783_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ).

% real_sqrt_ge_abs1
tff(fact_8784_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ).

% real_sqrt_ge_abs2
tff(fact_8785_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8786_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),numeral_numeral_real(bit0(one))) ) ) ).

% ln_sqrt
tff(fact_8787_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_Code_integer,K6: code_integer,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,P4)),K6)) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),K6))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8788_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_rat,K6: rat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,P4)),K6)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K6))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8789_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_int,K6: int,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K6)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8790_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_nat,K6: nat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K6)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8791_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: fun_nat_real,K6: real,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K6)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K6))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K6))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8792_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(bit0(one))) ).

% cos_30
tff(fact_8793_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(bit0(one))) ).

% sin_60
tff(fact_8794_arsinh__real__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_8795_complex__norm,axiom,
    ! [X: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(X),Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% complex_norm
tff(fact_8796_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8797_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8798_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M3))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8799_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pd(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8800_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N: nat] : minus_8373710615458151222nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(one_one_Code_integer,X)),groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_pe(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8801_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_pf(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8802_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] : aa_int_int(minus_minus_int(one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(times_times_int(aa_int_int(minus_minus_int(one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_pg(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8803_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ph(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% one_diff_power_eq'
tff(fact_8804_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8805_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_8806_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa4: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa4),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8807_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(bit0(one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8808_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_8809_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_8810_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pi(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_pj(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),groups6591440286371151544t_real(aTP_Lamm_pk(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_8811_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),one_one_real)) ).

% cos_x_y_le_one
tff(fact_8812_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(bit0(one))))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(bit0(one))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8813_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8814_cos__arctan,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% cos_arctan
tff(fact_8815_sin__arctan,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(arctan,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% sin_arctan
tff(fact_8816_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(bit0(one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(bit0(one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_8817_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ).

% sin_cos_sqrt
tff(fact_8818_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ).

% arctan_half
tff(fact_8819_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
     => ( groups4538972089207619220nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_int_int(times_times_int(N),aa_int_int(plus_plus_int(N),one_one_int))),aa_int_int(times_times_int(M),aa_int_int(minus_minus_int(M),one_one_int)))),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% Sum_Icc_int
tff(fact_8820_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_8821_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
     => ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ).

% cos_tan
tff(fact_8822_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ky(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8823_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_fg(F),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8824_sumr__cos__zero__one,axiom,
    ! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_pl,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8825_arcosh__def,axiom,
    ! [X: real] : aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arcosh_def
tff(fact_8826_freeze__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] : hoare_3904069481286416050_VEBTi(snga_assn_VEBT_VEBTi(A,Xs),array_8141364883501958055_VEBTi(A),aa_lis3169797078229555565i_assn(aTP_Lamm_pm(A),Xs)) ).

% freeze_rule
tff(fact_8827_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_8828_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_8829_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_8830_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = zero_zero_complex )
    <=> ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_8831_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% of_real_numeral
tff(fact_8832_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,numeral_numeral_real(W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_8833_of__real__mult,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_mult
tff(fact_8834_of__real__mult,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_mult
tff(fact_8835_of__real__divide,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_8836_of__real__divide,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_divide
tff(fact_8837_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_8838_of__real__add,axiom,
    ! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_8839_of__real__power,axiom,
    ! [X: real,N: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,X)),N) ).

% of_real_power
tff(fact_8840_of__real__power,axiom,
    ! [X: real,N: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,X)),N) ).

% of_real_power
tff(fact_8841_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8842_sin__of__real__pi,axiom,
    aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,pi)) = zero_zero_real ).

% sin_of_real_pi
tff(fact_8843_sin__of__real__pi,axiom,
    aa_complex_complex(sin_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_8844_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) ).

% of_real_neg_numeral
tff(fact_8845_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_8846_norm__of__real__add1,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_8847_norm__of__real__add1,axiom,
    ! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).

% norm_of_real_add1
tff(fact_8848_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),numeral_numeral_real(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_8849_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_8850_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_8851_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_8852_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_8853_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_8854_complex__of__real__def,axiom,
    ! [R2: real] : aa_real_complex(real_V4546457046886955230omplex,R2) = aa_real_complex(complex2(R2),zero_zero_real) ).

% complex_of_real_def
tff(fact_8855_complex__of__real__code,axiom,
    ! [X3: real] : aa_real_complex(real_V4546457046886955230omplex,X3) = aa_real_complex(complex2(X3),zero_zero_real) ).

% complex_of_real_code
tff(fact_8856_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa4: real] :
      ( ( aa_real_complex(complex2(X),Y) = aa_real_complex(real_V4546457046886955230omplex,Xa4) )
    <=> ( ( X = Xa4 )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_8857_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_8858_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_8859_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(X),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_8860_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(X),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_8861_complex__of__real__add__Complex,axiom,
    ! [R2: real,X: real,Y: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(X),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),X)),Y) ).

% complex_of_real_add_Complex
tff(fact_8862_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(X),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),R2)),Y) ).

% Complex_add_complex_of_real
tff(fact_8863_norm__less__p1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X))),one_one_real)))) ).

% norm_less_p1
tff(fact_8864_norm__less__p1,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex)))) ).

% norm_less_p1
tff(fact_8865_cos__int__times__real,axiom,
    ! [M: int,X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% cos_int_times_real
tff(fact_8866_cos__int__times__real,axiom,
    ! [M: int,X: real] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% cos_int_times_real
tff(fact_8867_sin__int__times__real,axiom,
    ! [M: int,X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% sin_int_times_real
tff(fact_8868_sin__int__times__real,axiom,
    ! [M: int,X: real] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).

% sin_int_times_real
tff(fact_8869_cos__sin__eq,axiom,
    ! [X: real] : aa_real_real(cos_real,X) = aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one)))),X)) ).

% cos_sin_eq
tff(fact_8870_cos__sin__eq,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,X) = aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one)))),X)) ).

% cos_sin_eq
tff(fact_8871_sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(sin_real,X) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one)))),X)) ).

% sin_cos_eq
tff(fact_8872_sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,X) = aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one)))),X)) ).

% sin_cos_eq
tff(fact_8873_minus__sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(bit0(one))))) ).

% minus_sin_cos_eq
tff(fact_8874_minus__sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one))))) ).

% minus_sin_cos_eq
tff(fact_8875_arsinh__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arsinh_def
tff(fact_8876_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pn,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8877_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pn,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8878_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pn,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8879_cos__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% cos_arcsin
tff(fact_8880_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_8881_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_8882_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_8883_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_8884_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_8885_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_8886_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8887_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8888_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8889_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8890_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8891_fact__Suc,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).

% fact_Suc
tff(fact_8892_fact__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).

% fact_Suc
tff(fact_8893_fact__Suc,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).

% fact_Suc
tff(fact_8894_fact__Suc,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_Suc
tff(fact_8895_fact__Suc,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).

% fact_Suc
tff(fact_8896_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,bit0(one))) = numeral_numeral_rat(bit0(one)) ).

% fact_2
tff(fact_8897_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% fact_2
tff(fact_8898_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,bit0(one))) = numeral_numeral_real(bit0(one)) ).

% fact_2
tff(fact_8899_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% fact_2
tff(fact_8900_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,bit0(one))) = numera6690914467698888265omplex(bit0(one)) ).

% fact_2
tff(fact_8901_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) ).

% arcsin_1
tff(fact_8902_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ).

% arcsin_minus_1
tff(fact_8903_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) != zero_zero_rat ).

% fact_nonzero
tff(fact_8904_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) != zero_zero_int ).

% fact_nonzero
tff(fact_8905_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N) != zero_z3403309356797280102nteger ).

% fact_nonzero
tff(fact_8906_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) != zero_zero_real ).

% fact_nonzero
tff(fact_8907_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) != zero_zero_nat ).

% fact_nonzero
tff(fact_8908_fact__nonzero,axiom,
    ! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) != zero_zero_complex ).

% fact_nonzero
tff(fact_8909_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_ge_zero
tff(fact_8910_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ).

% fact_ge_zero
tff(fact_8911_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_ge_zero
tff(fact_8912_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_ge_zero
tff(fact_8913_fact__ge__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_zero
tff(fact_8914_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat)) ).

% fact_not_neg
tff(fact_8915_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).

% fact_not_neg
tff(fact_8916_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),zero_z3403309356797280102nteger)) ).

% fact_not_neg
tff(fact_8917_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N)),zero_zero_real)) ).

% fact_not_neg
tff(fact_8918_fact__not__neg,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).

% fact_not_neg
tff(fact_8919_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).

% fact_gt_zero
tff(fact_8920_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).

% fact_gt_zero
tff(fact_8921_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ).

% fact_gt_zero
tff(fact_8922_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).

% fact_gt_zero
tff(fact_8923_fact__gt__zero,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_gt_zero
tff(fact_8924_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ) ).

% fact_less_mono
tff(fact_8925_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).

% fact_less_mono
tff(fact_8926_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ) ) ).

% fact_less_mono
tff(fact_8927_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ) ).

% fact_less_mono
tff(fact_8928_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono
tff(fact_8929_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8930_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,N),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_8931_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8932_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_8933_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_8934_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_8935_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).

% fact_fact_dvd_fact
tff(fact_8936_fact__le__power,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_8937_fact__le__power,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_8938_fact__le__power,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).

% fact_le_power
tff(fact_8939_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).

% choose_dvd
tff(fact_8940_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).

% choose_dvd
tff(fact_8941_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).

% choose_dvd
tff(fact_8942_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).

% choose_dvd
tff(fact_8943_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8944_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8945_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8946_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8947_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8948_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8949_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_8950_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% square_fact_le_2_fact
tff(fact_8951_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_8952_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_8953_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_8954_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_8955_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_8956_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8957_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8958_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8959_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8960_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8961_fact__reduce,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_complex(semiri5044797733671781792omplex,N) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8962_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_po(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8963_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_pp(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8964_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_pq(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8965_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_pr(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8966_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_ps(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8967_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: fun_na5509613867492920147r_real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun5144707104869965066t_real(aTP_Lamm_pt(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_Code_integer_real(aa_nat8538992517141809556r_real(Diff,zero_zero_nat),zero_z3403309356797280102nteger) ) ) ) ).

% Maclaurin_zero
tff(fact_8968_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J2: fun_nat_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pu(H),J2),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_8969_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8970_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ) ).

% arcsin_bounded
tff(fact_8971_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ).

% arcsin_ubound
tff(fact_8972_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_8973_arcsin__sin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_8974_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_8975_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),pi))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_8976_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),Y))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sin_real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8977_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(arcsin,X)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8978_cos__coeff__def,axiom,
    ! [X3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X3))
       => ( aa_nat_real(cos_coeff,X3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X3)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X3))
       => ( aa_nat_real(cos_coeff,X3) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_8979_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pv,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8980_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T3))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pv,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8981_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pv,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8982_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] : aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pv,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ).

% Maclaurin_sin_expansion
tff(fact_8983_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8984_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_8985_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_8986_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_8987_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_8988_sin__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% sin_coeff_Suc
tff(fact_8989_cos__coeff__Suc,axiom,
    ! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).

% cos_coeff_Suc
tff(fact_8990_sin__coeff__def,axiom,
    ! [X3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X3))
       => ( aa_nat_real(sin_coeff,X3) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),X3))
       => ( aa_nat_real(sin_coeff,X3) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X3)) ) ) ) ).

% sin_coeff_def
tff(fact_8991_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pw,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8992_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_px,X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_8993_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ).

% sin_arccos_abs
tff(fact_8994_sin__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) ) ) ).

% sin_arccos
tff(fact_8995_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))) ) ).

% exp_less_mono
tff(fact_8996_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_8997_sums__zero,axiom,
    pp(aa_complex_bool(sums_complex(aTP_Lamm_dx),zero_zero_complex)) ).

% sums_zero
tff(fact_8998_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_dy),zero_zero_real)) ).

% sums_zero
tff(fact_8999_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_am),zero_zero_nat)) ).

% sums_zero
tff(fact_9000_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_dz),zero_zero_int)) ).

% sums_zero
tff(fact_9001_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_9002_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_9003_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,X) = one_one_real )
    <=> ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_9004_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9005_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_less_exp_iff
tff(fact_9006_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_9007_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_9008_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% one_le_exp_iff
tff(fact_9009_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% exp_ln_iff
tff(fact_9010_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_9011_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_cp(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_9012_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_cq(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_9013_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) ).

% arccos_0
tff(fact_9014_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_el(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_9015_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel
tff(fact_9016_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_9017_exp__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_gt_zero
tff(fact_9018_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X2: real] : aa_real_real(exp_real,X2) = Y ) ).

% exp_total
tff(fact_9019_exp__ge__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_ge_zero
tff(fact_9020_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_9021_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).

% exp_gt_one
tff(fact_9022_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ).

% exp_ge_add_one_self
tff(fact_9023_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_9024_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real)))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_9025_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_9026_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_9027_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_9028_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_9029_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_9030_exp__le,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one)))) ).

% exp_le
tff(fact_9031_arccos__cos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_9032_exp__half__le2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),numeral_numeral_real(bit0(one)))) ).

% exp_half_le2
tff(fact_9033_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_9034_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_9035_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9036_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_9037_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_py),one_one_real)) ).

% power_half_series
tff(fact_9038_arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9039_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => pp(aa_real_bool(sums_real(aTP_Lamm_pz(G)),X)) ) ).

% sums_if'
tff(fact_9040_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => ( pp(aa_real_bool(sums_real(F),Y))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qa(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sums_if
tff(fact_9041_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ).

% exp_bound
tff(fact_9042_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_9043_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_9044_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_9045_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ).

% arccos_le_pi2
tff(fact_9046_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
      & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pw,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_9047_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),numeral_numeral_real(bit0(one))))),aa_real_real(exp_real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9048_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_9049_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one)))),X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one)))),X)))) ).

% tanh_real_altdef
tff(fact_9050_cos__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_qb,X)),aa_real_real(cos_real,X))) ).

% cos_paired
tff(fact_9051_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_9052_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9053_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf(A,B)) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_9054_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K = zero_zero_int )
           => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = K ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,bit0(one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_9055_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_9056_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_9057_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(2)
tff(fact_9058_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(6)
tff(fact_9059_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(X),Y))) ) ) ).

% OR_lower
tff(fact_9060_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L))) ) ).

% or_greater_eq
tff(fact_9061_plus__and__or,axiom,
    ! [X: int,Y: int] : aa_int_int(plus_plus_int(aa_int_int(bit_se725231765392027082nd_int(X),Y)),aa_int_int(bit_se1409905431419307370or_int(X),Y)) = aa_int_int(plus_plus_int(X),Y) ).

% plus_and_or
tff(fact_9062_prod__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_9063_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_9064_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),groups708209901874060359at_nat(aTP_Lamm_ar,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_9065_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_ar,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_9066_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ) ) ).

% OR_upper
tff(fact_9067_or__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% or_int_rec
tff(fact_9068_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9069_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9070_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_9071_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_9072_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,bit0(Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9073_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_9074_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_9075_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_9076_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_9077_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_9078_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq
tff(fact_9079_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_9080_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9081_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9082_prod__int__plus__eq,axiom,
    ! [I: nat,J2: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2))) = groups1705073143266064639nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)))) ).

% prod_int_plus_eq
tff(fact_9083_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).

% floor_eq2
tff(fact_9084_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_9085_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% or_Suc_0_eq
tff(fact_9086_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% Suc_0_or_eq
tff(fact_9087_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% or_nat_rec
tff(fact_9088_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_9089_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_9090_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9091_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),B))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9092_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_9093_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,X)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pv,X),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_9094_binomial__Suc__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).

% binomial_Suc_n
tff(fact_9095_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_9096_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_9097_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_9098_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_9099_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_9100_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_9101_real__sqrt__inverse,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ).

% real_sqrt_inverse
tff(fact_9102_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_9103_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).

% Suc_times_binomial
tff(fact_9104_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_9105_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_9106_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R2))) ) ).

% binomial_le_pow
tff(fact_9107_divide__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_9108_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).

% zero_less_binomial
tff(fact_9109_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_9110_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_9111_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_9112_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_9113_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).

% inverse_powr
tff(fact_9114_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_9115_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_9116_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D3: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D3),E2))
         => ( pp(aa_real_bool(P,D3))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono_1
tff(fact_9117_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E: real] :
      ( ! [D3: real,E2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D3),E2))
         => ( pp(aa_real_bool(P,D3))
           => pp(aa_real_bool(P,E2)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(P,E)) ) ) ) ).

% forall_pos_mono
tff(fact_9118_real__arch__inverse,axiom,
    ! [E: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
    <=> ? [N7: nat] :
          ( ( N7 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N7))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N7))),E)) ) ) ).

% real_arch_inverse
tff(fact_9119_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9120_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_9121_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% binomial_maximum
tff(fact_9122_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_antimono
tff(fact_9123_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K4)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4))) ) ) ).

% binomial_mono
tff(fact_9124_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),N))) ).

% binomial_maximum'
tff(fact_9125_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% binomial_le_pow2
tff(fact_9126_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_9127_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_9128_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_9129_log__inverse,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) ) ) ) ) ).

% log_inverse
tff(fact_9130_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_9131_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K4)),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4))) ) ) ).

% binomial_strict_mono
tff(fact_9132_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_9133_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% central_binomial_odd
tff(fact_9134_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_9135_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X))))) ).

% exp_plus_inverse_exp
tff(fact_9136_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% choose_two
tff(fact_9137_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_9138_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9139_tan__cot,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_9140_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(bit0(one))))) ) ).

% real_le_x_sinh
tff(fact_9141_real__le__abs__sinh,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(bit0(one)))))) ).

% real_le_abs_sinh
tff(fact_9142_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(bit0(one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_9143_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_9144_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_9145_real__scaleR__def,axiom,
    ! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),X) ).

% real_scaleR_def
tff(fact_9146_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_9147_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_9148_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_9149_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).

% complex_scaleR
tff(fact_9150_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9151_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qc,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_9152_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qd,R2)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),N))),N) ).

% sum_choose_lower
tff(fact_9153_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qe,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9154_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qe,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_9155_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qf(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_9156_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qg(M),N),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R2) ).

% vandermonde
tff(fact_9157_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9158_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% choose_row_sum
tff(fact_9159_binomial,axiom,
    ! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qh(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_9160_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I3))
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J))
           => ( aa_nat_nat(B,J) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qk(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_9161_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),N) ).

% choose_square_sum
tff(fact_9162_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% complex_inverse
tff(fact_9163_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M)) ).

% binomial_r_part_sum
tff(fact_9164_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_9165_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sinh_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9166_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% sinh_real_less_iff
tff(fact_9167_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_9168_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_9169_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% sinh_real_pos_iff
tff(fact_9170_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_9171_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_9172_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_less_cosh_real
tff(fact_9173_cosh__real__nonzero,axiom,
    ! [X: real] : aa_real_real(cosh_real,X) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9174_cosh__real__pos,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_pos
tff(fact_9175_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9176_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9177_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_nonneg
tff(fact_9178_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_9179_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9180_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9181_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_9182_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(bit0(one))) ) ) ).

% cosh_ln_real
tff(fact_9183_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(bit0(one))) ) ) ).

% sinh_ln_real
tff(fact_9184_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(bit0(one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9185_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_9186_divide__i,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).

% divide_i
tff(fact_9187_divide__numeral__i,axiom,
    ! [Z: complex,N: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(N)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(N)) ).

% divide_numeral_i
tff(fact_9188_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9189_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_9190_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9191_less__eq__mask,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% less_eq_mask
tff(fact_9192_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_nonnegative_int
tff(fact_9193_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_9194_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_9195_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_complex(complex2(X),Y) = imaginary_unit )
    <=> ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9196_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9197_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9198_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9199_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% Suc_mask_eq_exp
tff(fact_9200_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% mask_nat_less_exp
tff(fact_9201_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_9202_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% mask_half_int
tff(fact_9203_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),one_one_int) ).

% mask_int_def
tff(fact_9204_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_9205_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R: real,A4: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A4))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A4))))) ).

% complex_split_polar
tff(fact_9206_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_9207_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9208_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(bit0(one))))) ).

% csqrt_ii
tff(fact_9209_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one))) ).

% Arg_minus_ii
tff(fact_9210_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) ).

% Arg_ii
tff(fact_9211_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,bit0(one))) = Z ).

% power2_csqrt
tff(fact_9212_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_9213_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).

% of_real_sqrt
tff(fact_9214_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ).

% Arg_bounded
tff(fact_9215_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9216_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_9217_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9218_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9219_cot__npi,axiom,
    ! [N: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9220_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9221_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9222_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_9223_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_9224_cis__divide,axiom,
    ! [A: real,B: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% cis_divide
tff(fact_9225_DeMoivre,axiom,
    ! [A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).

% DeMoivre
tff(fact_9226_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X))) ) ) ).

% cot_gt_zero
tff(fact_9227_tan__cot_H,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_9228_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => bij_betw_nat_complex(aTP_Lamm_qn(N),aa_nat_set_nat(set_ord_lessThan_nat,N),collect_complex(aTP_Lamm_dv(N))) ) ).

% bij_betw_roots_unity
tff(fact_9229_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(bitM(M),bit0(one)) = product_Pair_int_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9230_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,bit0(K)) = aa_num_nat(numeral_numeral_nat,bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_9231_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_9232_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa_num_num(bit1,N)) = aa_num_num(bit1,bit0(N)) ).

% semiring_norm(28)
tff(fact_9233_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(bit0(N)) = aa_num_num(bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_9234_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,bit0(N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_9235_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bitM(N)),one) = bit0(N) ).

% BitM_plus_one
tff(fact_9236_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bitM(N)) = bit0(N) ).

% one_plus_BitM
tff(fact_9237_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_9238_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_9239_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,inc(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_9240_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_9241_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bit0(N)))) ).

% or_minus_numerals(8)
tff(fact_9242_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bit0(N)))) ).

% or_minus_numerals(4)
tff(fact_9243_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_9244_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_9245_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_9246_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_9247_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9248_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(minus_minus_int(K),L)) ).

% take_bit_diff
tff(fact_9249_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(times_times_int(K),L)) ).

% take_bit_mult
tff(fact_9250_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),inc(Y)) = inc(aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) ).

% add_inc
tff(fact_9251_num__induct,axiom,
    ! [P: fun_num_bool,X: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X2: num] :
            ( pp(aa_num_bool(P,X2))
           => pp(aa_num_bool(P,inc(X2))) )
       => pp(aa_num_bool(P,X)) ) ) ).

% num_induct
tff(fact_9252_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9253_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9254_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9255_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N),K))) ).

% take_bit_nonnegative
tff(fact_9256_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_9257_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_9258_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,bit0(N)),one) = bit0(one) ).

% or_not_num_neg.simps(4)
tff(fact_9259_inc_Osimps_I1_J,axiom,
    inc(one) = bit0(one) ).

% inc.simps(1)
tff(fact_9260_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,bit0(N)),aa_num_num(bit1,M)) = bit0(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9261_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9262_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9263_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_9264_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa_num_num(bit1,X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_9265_add__One,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),one) = inc(X) ).

% add_One
tff(fact_9266_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,bit0(N)),bit0(M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9267_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = bit0(N) ).

% inc_BitM_eq
tff(fact_9268_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit1,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(9)
tff(fact_9269_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_9270_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,X),inc(Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)),X) ).

% mult_inc
tff(fact_9271_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(minus_minus_int(K),one_one_int)) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9272_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bit0(M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9273_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),bit0(M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9274_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(plus_plus_int(K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9275_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9276_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))) ).

% take_bit_nat_less_exp
tff(fact_9277_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9278_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% take_bit_nat_def
tff(fact_9279_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9280_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa4: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X),Xa4) = Y )
     => ( ( ( X = one )
         => ( ( Xa4 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M3: num] :
                ( ( Xa4 = bit0(M3) )
               => ( Y != aa_num_num(bit1,M3) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa4 = aa_num_num(bit1,M3) )
                 => ( Y != aa_num_num(bit1,M3) ) ) )
           => ( ( ? [N2: num] : X = bit0(N2)
               => ( ( Xa4 = one )
                 => ( Y != bit0(one) ) ) )
             => ( ! [N2: num] :
                    ( ( X = bit0(N2) )
                   => ! [M3: num] :
                        ( ( Xa4 = bit0(M3) )
                       => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = bit0(N2) )
                     => ! [M3: num] :
                          ( ( Xa4 = aa_num_num(bit1,M3) )
                         => ( Y != bit0(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
                 => ( ( ? [N2: num] : X = aa_num_num(bit1,N2)
                     => ( ( Xa4 = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa_num_num(bit1,N2) )
                         => ! [M3: num] :
                              ( ( Xa4 = bit0(M3) )
                             => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X = aa_num_num(bit1,N2) )
                           => ! [M3: num] :
                                ( ( Xa4 = aa_num_num(bit1,M3) )
                               => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9281_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ).

% take_bit_int_less_exp
tff(fact_9282_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(modulo_modulo_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ).

% take_bit_int_def
tff(fact_9283_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9284_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_9285_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9286_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_9287_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9288_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9289_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9290_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(plus_plus_int(K),one_one_int)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_9291_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9292_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_9293_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_9294_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9295_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),aa_int_int(plus_plus_int(K),one_one_int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9296_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9297_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aTP_Lamm_dv(N)),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qo(C),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9298_real__root__zero,axiom,
    ! [N: nat] : aa_real_real(aa_nat_fun_real_real(root,N),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9299_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_9300_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9301_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_9302_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9303_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_9304_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_9305_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9306_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9307_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_9308_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_9309_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_9310_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_9311_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_9312_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_9313_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_9314_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_9315_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_9316_real__root__minus,axiom,
    ! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_minus
tff(fact_9317_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_nat_fun_real_real(root,M),X)) ).

% real_root_commute
tff(fact_9318_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),X) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_mult_exp
tff(fact_9319_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ).

% real_root_mult
tff(fact_9320_real__root__divide,axiom,
    ! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ).

% real_root_divide
tff(fact_9321_real__root__inverse,axiom,
    ! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).

% real_root_inverse
tff(fact_9322_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ).

% real_root_pos_pos_le
tff(fact_9323_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_less_mono
tff(fact_9324_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).

% real_root_le_mono
tff(fact_9325_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).

% real_root_power
tff(fact_9326_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).

% real_root_abs
tff(fact_9327_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_gt_zero
tff(fact_9328_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9329_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% sqrt_def
tff(fact_9330_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9331_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).

% real_root_pos_pos
tff(fact_9332_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9333_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N4),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_9334_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ).

% odd_real_root_pow
tff(fact_9335_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9336_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_9337_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_9338_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_9339_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9340_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N4),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_9341_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_9342_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% log_base_root
tff(fact_9343_ln__root,axiom,
    ! [N: nat,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_9344_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_9345_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_qp(L),Qr) ).

% divmod_step_nat_def
tff(fact_9346_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_qq(L),Qr) ).

% divmod_step_int_def
tff(fact_9347_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_qr(L),Qr) ).

% divmod_step_integer_def
tff(fact_9348_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
       => ( divmod_nat(M,N) = produc2626176000494625587at_nat(aTP_Lamm_qs,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_9349_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).

% divmod_integer'_def
tff(fact_9350_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9351_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9352_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_qt,Qr) ).

% Divides.adjust_div_def
tff(fact_9353_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% divmod_nat_def
tff(fact_9354_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9355_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9356_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = uminus1351360451143612070nteger(aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9357_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_qu(D))) ).

% int_ge_less_than_def
tff(fact_9358_divide__integer_Oabs__eq,axiom,
    ! [Xa4: int,X: int] : divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xa4),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa4),X)) ).

% divide_integer.abs_eq
tff(fact_9359_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa4: nat,X: int] : bit_se8260200283734997820nteger(Xa4,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(bit_se4203085406695923979it_int(Xa4),X)) ).

% unset_bit_integer.abs_eq
tff(fact_9360_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9361_less__integer_Oabs__eq,axiom,
    ! [Xa4: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa4)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa4),X)) ) ).

% less_integer.abs_eq
tff(fact_9362_plus__integer_Oabs__eq,axiom,
    ! [Xa4: int,X: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa4),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(plus_plus_int(Xa4),X)) ).

% plus_integer.abs_eq
tff(fact_9363_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_qv(D))) ).

% int_ge_less_than2_def
tff(fact_9364_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_9365_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_qw,Y)) ).

% arctan_def
tff(fact_9366_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_qx,Y)) ).

% arcsin_def
tff(fact_9367_nat__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N)) = N ).

% nat_int
tff(fact_9368_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_9369_nat__of__bool,axiom,
    ! [P: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_nat(zero_n2687167440665602831ol_nat,P) ).

% nat_of_bool
tff(fact_9370_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9371_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9372_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_9373_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_9374_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% zless_nat_conj
tff(fact_9375_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9376_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_9377_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_9378_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_9379_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9380_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9381_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_abs_dvd_iff
tff(fact_9382_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_9383_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_9384_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_9385_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_9386_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9387_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9388_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9389_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9390_nat__numeral__as__int,axiom,
    ! [X3: num] : aa_num_nat(numeral_numeral_nat,X3) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X3)) ).

% nat_numeral_as_int
tff(fact_9391_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9392_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_9393_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X5))) ) ) ).

% ex_nat
tff(fact_9394_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ! [X5: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X5))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X5))) ) ) ).

% all_nat
tff(fact_9395_eq__nat__nat__iff,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z3))
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z3) )
        <=> ( Z = Z3 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9396_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% unset_bit_nat_def
tff(fact_9397_nat__mask__eq,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% nat_mask_eq
tff(fact_9398_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_9399_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_9400_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).

% nat_le_iff
tff(fact_9401_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_9402_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ) ).

% int_eq_iff
tff(fact_9403_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9404_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(times_times_int(W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9405_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))))) ).

% real_nat_ceiling_ge
tff(fact_9406_nat__plus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_9407_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% and_nat_def
tff(fact_9408_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% or_nat_def
tff(fact_9409_nat__times__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_9410_nat__div__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_9411_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_qy) ) ) ).

% ln_neg_is_const
tff(fact_9412_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_9413_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_9414_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( W = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_9415_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N7: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N7) )
           => pp(aa_nat_bool(P,N7)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_9416_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_9417_nat__add__distrib,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z3))
       => ( aa_int_nat(nat2,aa_int_int(plus_plus_int(Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) ) ) ) ).

% nat_add_distrib
tff(fact_9418_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).

% le_nat_iff
tff(fact_9419_Suc__as__int,axiom,
    ! [X3: nat] : aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ).

% Suc_as_int
tff(fact_9420_nat__mult__distrib,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) ) ) ).

% nat_mult_distrib
tff(fact_9421_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9422_nat__diff__distrib,axiom,
    ! [Z3: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z3))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z3),Z))
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) ) ) ) ).

% nat_diff_distrib
tff(fact_9423_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_9424_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9425_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9426_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_9427_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(modulo_modulo_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9428_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9429_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9430_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_9431_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9432_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9433_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_9434_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_9435_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_9436_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_9437_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_9438_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z3))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9439_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_9440_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_9441_diff__nat__eq__if,axiom,
    ! [Z3: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) = aa_int_nat(nat2,Z) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(minus_minus_int(Z),Z3)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z3))) ) ) ) ).

% diff_nat_eq_if
tff(fact_9442_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_qz,Y)) ).

% arccos_def
tff(fact_9443_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_9444_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)) ) ) ).

% even_nat_iff
tff(fact_9445_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) = the_real(aTP_Lamm_ra) ).

% pi_half
tff(fact_9446_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),the_real(aTP_Lamm_ra)) ).

% pi_def
tff(fact_9447_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,N)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_9448_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_9449_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_9450_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R2)),L)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9451_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(times_times_int(L),aa_int_int(sgn_sgn_int,R2))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9452_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R2)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9453_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(times_times_int(K),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9454_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_9455_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_9456_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9457_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_9458_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9459_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9460_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q2),L)),R2) )
         => pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),product_Pair_int_int(Q2,R2))) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9461_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9462_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(times_times_int(Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4)))
            & ( K3 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9463_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(times_times_int(Q3),A22) ) ) )
         => ~ ! [R: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R) )
               => ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q3),A22)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9464_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_9465_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_9466_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(times_times_int(A),B)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_9467_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9468_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_sgn_iff
tff(fact_9469_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_9470_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_9471_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_9472_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N),K)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9473_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_9474_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9475_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9476_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_9477_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,N)))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_9478_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,bit0(W))),N))
      <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9479_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),N))
      <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9480_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9481_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9482_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),N))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(minus_minus_int(K),one_one_int))),N)) ) ).

% bit_minus_int_iff
tff(fact_9483_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),N))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% bit_not_int_iff
tff(fact_9484_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),N))
    <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),N)) ) ) ).

% bit_and_int_iff
tff(fact_9485_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),N))
    <=> ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
        | pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),N)) ) ) ).

% bit_or_int_iff
tff(fact_9486_real__sgn__eq,axiom,
    ! [X: real] : aa_real_real(sgn_sgn_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(abs_abs_real,X)) ).

% real_sgn_eq
tff(fact_9487_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9488_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_9489_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9490_and__not__numerals_I1_J,axiom,
    aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9491_or__not__numerals_I1_J,axiom,
    aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9492_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
    <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ).

% bit_not_int_iff'
tff(fact_9493_sgn__eq,axiom,
    ! [Z: complex] : aa_complex_complex(sgn_sgn_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% sgn_eq
tff(fact_9494_sgn__real__def,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_9495_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% not_int_div_2
tff(fact_9496_even__not__iff__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)) ) ).

% even_not_iff_int
tff(fact_9497_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9498_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,bit0(M)) ).

% and_not_numerals(4)
tff(fact_9499_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9500_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9501_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N))) ).

% or_not_numerals(2)
tff(fact_9502_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N))) ).

% int_numeral_or_not_num_neg
tff(fact_9503_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9504_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% numeral_or_not_num_eq
tff(fact_9505_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),N)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9506_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),M2))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_9507_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(5)
tff(fact_9508_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,bit0(M)) ).

% and_not_numerals(7)
tff(fact_9509_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N))) ).

% or_not_numerals(3)
tff(fact_9510_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9511_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9512_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(6)
tff(fact_9513_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(9)
tff(fact_9514_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% or_not_numerals(6)
tff(fact_9515_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_9516_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_9517_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))) ) ).

% bit_int_def
tff(fact_9518_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => ( aa_complex_real(arg,Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_9519_split__root,axiom,
    ! [P: fun_real_bool,N: nat,X: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ! [Y6: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y6)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y6)),N)) = X )
             => pp(aa_real_bool(P,Y6)) ) ) ) ) ).

% split_root
tff(fact_9520_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_9521_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ) ).

% Arg_correct
tff(fact_9522_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% and_not_numerals(8)
tff(fact_9523_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_9524_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(N)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_9525_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% not_int_rec
tff(fact_9526_Bit__Operations_Oset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(plus_plus_int(K),aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ).

% Bit_Operations.set_bit_eq
tff(fact_9527_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(minus_minus_int(K),aa_int_int(times_times_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ).

% unset_bit_eq
tff(fact_9528_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_Suc_from_most
tff(fact_9529_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),pi)),numeral_numeral_real(bit0(one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_9530_int__not__code_I1_J,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_9531_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_9532_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9533_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_9534_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_9535_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9536_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),N))
    <=> ~ ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N))
        <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_9537_int__xor__code_I1_J,axiom,
    ! [J2: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),J2) = J2 ).

% int_xor_code(1)
tff(fact_9538_int__xor__code_I2_J,axiom,
    ! [I: int] : aa_int_int(bit_se6526347334894502574or_int(I),zero_zero_int) = I ).

% int_xor_code(2)
tff(fact_9539_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(X),Y))) ) ) ).

% XOR_lower
tff(fact_9540_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).

% not_bit_Suc_0_numeral
tff(fact_9541_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9542_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_9543_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ) ).

% bit_nat_def
tff(fact_9544_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N))) ) ) ) ).

% XOR_upper
tff(fact_9545_test__bit__int__code_I1_J,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,zero_zero_int),N)) ).

% test_bit_int_code(1)
tff(fact_9546_int__and__code_I2_J,axiom,
    ! [I: int] : aa_int_int(bit_se725231765392027082nd_int(I),zero_zero_int) = zero_zero_int ).

% int_and_code(2)
tff(fact_9547_int__and__code_I1_J,axiom,
    ! [J2: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),J2) = zero_zero_int ).

% int_and_code(1)
tff(fact_9548_int__or__code_I1_J,axiom,
    ! [J2: int] : aa_int_int(bit_se1409905431419307370or_int(zero_zero_int),J2) = J2 ).

% int_or_code(1)
tff(fact_9549_int__or__code_I2_J,axiom,
    ! [I: int] : aa_int_int(bit_se1409905431419307370or_int(I),zero_zero_int) = I ).

% int_or_code(2)
tff(fact_9550_xor__int__rec,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% xor_int_rec
tff(fact_9551_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( aa_complex_real(arg,Z) = zero_zero_real ) )
      & ( ( Z != zero_zero_complex )
       => ( aa_complex_real(arg,Z) = fChoice_real(aTP_Lamm_rb(Z)) ) ) ) ).

% Arg_def
tff(fact_9552_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_9553_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_9554_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_9555_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,bit0(Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9556_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% xor_nat_def
tff(fact_9557_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
          & ( ( N != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_9558_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_bool_bool(fequal_bool(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% xor_nat_rec
tff(fact_9559_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% Suc_0_xor_eq
tff(fact_9560_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),aa_bool_nat(zero_n2687167440665602831ol_nat,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))) ).

% xor_Suc_0_eq
tff(fact_9561_int__lsb__numeral_I1_J,axiom,
    ~ pp(aa_int_bool(least_4859182151741483524sb_int,zero_zero_int)) ).

% int_lsb_numeral(1)
tff(fact_9562_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ pp(aa_int_bool(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,bit0(W)))) ).

% int_lsb_numeral(6)
tff(fact_9563_int__lsb__numeral_I3_J,axiom,
    pp(aa_int_bool(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(3)
tff(fact_9564_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : pp(aa_int_bool(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))) ).

% int_lsb_numeral(7)
tff(fact_9565_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ pp(aa_int_bool(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W))))) ).

% int_lsb_numeral(8)
tff(fact_9566_int__lsb__numeral_I5_J,axiom,
    pp(aa_int_bool(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)))) ).

% int_lsb_numeral(5)
tff(fact_9567_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : pp(aa_int_bool(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))))) ).

% int_lsb_numeral(9)
tff(fact_9568_lsb__int__def,axiom,
    ! [I: int] :
      ( pp(aa_int_bool(least_4859182151741483524sb_int,I))
    <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,I),zero_zero_nat)) ) ).

% lsb_int_def
tff(fact_9569_bin__last__conv__lsb,axiom,
    aTP_Lamm_rc = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9570_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one)))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_9571_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_9572_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_9573_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat)))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,bit0(one))))),one_one_nat))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_9574_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,bit0(one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,bit0(one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_9575_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_9576_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N2)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N2))
                       => ( Y = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,bit0(one))))))))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N2)))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_9577_vebt__minti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(X) = Y )
     => ( pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leafi(A4,B4) )
             => ( ( ( pp(A4)
                   => ( Y = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                  & ( ~ pp(A4)
                   => ( ( pp(B4)
                       => ( Y = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                      & ( ~ pp(B4)
                       => ( Y = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) )
               => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Leafi(A4,B4))) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) )
                 => ( ( Y = heap_T3487192422709364219on_nat(some_nat(Mi2)) )
                   => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz))) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_9578_vebt__maxti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(X) = Y )
     => ( pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leafi(A4,B4) )
             => ( ( ( pp(B4)
                   => ( Y = heap_T3487192422709364219on_nat(some_nat(one_one_nat)) ) )
                  & ( ~ pp(B4)
                   => ( ( pp(A4)
                       => ( Y = heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)) ) )
                      & ( ~ pp(A4)
                       => ( Y = heap_T3487192422709364219on_nat(none_nat) ) ) ) ) )
               => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Leafi(A4,B4))) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv),Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( X = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) )
                 => ( ( Y = heap_T3487192422709364219on_nat(some_nat(Ma2)) )
                   => ~ pp(aa_VEBT_VEBTi_bool(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz))) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_9579_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9580_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),pi)) = zero_zero_real )
    <=> pp(aa_set_real_bool(member_real(X),ring_1_Ints_real)) ) ).

% sin_times_pi_eq_0
tff(fact_9581_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9582_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9583_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg,Tree_list2),Summary)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2)) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,Mmoi2),Mmo2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Degi2),Deg)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_rd(Tree_list2),Tree_array2))) ).

% vebt_assn_raw.simps(2)
tff(fact_9584_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Leaf(A4,B4))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9585_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Leaf(A4,B4))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9586_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = one_one_real )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),vEBT_Leaf(A4,B4))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_9587_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = one_one_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Leaf(A4,B4))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_9588_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9589_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_9590_drop__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9591_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9592_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9593_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9594_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9595_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9596_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_9597_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se8568078237143864401it_int(zero_zero_nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_9598_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(N),K)) ).

% drop_bit_nat_eq
tff(fact_9599_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(bit_se8568078237143864401it_int(one_one_nat),I) ).

% bin_rest_code
tff(fact_9600_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(N),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ).

% drop_bit_int_def
tff(fact_9601_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% drop_bit_nat_def
tff(fact_9602_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( pp(B4)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B4)
                   => ( ( pp(A4)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A4)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf(A4,B4))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9603_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( pp(A4)
                   => ( Y = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A4)
                   => ( ( pp(B4)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B4)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A4,B4))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9604_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A4),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf(A4,B4))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9605_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B4),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf(A4,B4))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz))) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9606_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( ( Y = one_one_nat )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv) )
               => ( ( Y = one_one_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf(fTrue,Uv))) ) )
           => ( ! [Uu: bool] :
                  ( ( X = vEBT_Leaf(Uu,fTrue) )
                 => ( ( Y = one_one_nat )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf(Uu,fTrue))) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
                   => ( ( Y = one_one_nat )
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy))) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2) )
                     => ( ( Y = one_one_nat )
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2))) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9607_merge__true__star,axiom,
    aa_assn_assn(times_times_assn(top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_9608_ent__true,axiom,
    ! [P: assn] : entails(P,top_top_assn) ).

% ent_true
tff(fact_9609_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(top_top_assn),aa_assn_assn(times_times_assn(top_top_assn),P)) = aa_assn_assn(times_times_assn(top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_9610_ent__star__mono__true,axiom,
    ! [A2: assn,A5: assn,B2: assn,B8: assn] :
      ( entails(A2,aa_assn_assn(times_times_assn(A5),top_top_assn))
     => ( entails(B2,aa_assn_assn(times_times_assn(B8),top_top_assn))
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B2)),top_top_assn),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A5),B8)),top_top_assn)) ) ) ).

% ent_star_mono_true
tff(fact_9611_ent__refl__true,axiom,
    ! [A2: assn] : entails(A2,aa_assn_assn(times_times_assn(A2),top_top_assn)) ).

% ent_refl_true
tff(fact_9612_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R3: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(Q),top_top_assn))
     => entails(aa_assn_assn(times_times_assn(P),R3),aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% ent_true_drop(1)
tff(fact_9613_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% ent_true_drop(2)
tff(fact_9614_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( rep_assn(P,H)
     => rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn),H) ) ).

% mod_star_trueI
tff(fact_9615_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn),H)
     => ~ ! [H4: produc3658429121746597890et_nat] : ~ rep_assn(P,H4) ) ).

% mod_star_trueE
tff(fact_9616_shiftr__integer__conv__div__pow2,axiom,
    ! [N: nat,X: code_integer] : bit_se3928097537394005634nteger(N,X) = divide6298287555418463151nteger(X,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9617_vebt__assn__raw_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa4: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,X),Xa4) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ! [Ai: bool,Bi: bool] :
                ( ( Xa4 = vEBT_Leafi(Ai,Bi) )
               => ( Y != aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_bool_bool(fequal_bool(Ai),A4)),aa_bool_bool(fequal_bool(Bi),B4))) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xa4 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                 => ( Y != aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,Mmoi),Mmo)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Degi),Deg2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_rd(Tree_list),Tree_array))) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va,Vb3),Vc3)
             => ( ? [Vd3: bool,Ve3: bool] : Xa4 = vEBT_Leafi(Vd3,Ve3)
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: bool,Ve3: bool] : X = vEBT_Leaf(Vd3,Ve3)
               => ( ? [V2: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xa4 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va,Vb3),Vc3)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_9618_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv) )
               => ( ~ pp(Y)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv))) ) )
           => ( ! [Uu: bool] :
                  ( ( X = vEBT_Leaf(Uu,fTrue) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu,fTrue))) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy))) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9619_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_9620_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_9621_ent__false__iff,axiom,
    ! [P: assn] :
      ( entails(P,bot_bot_assn)
    <=> ! [H2: produc3658429121746597890et_nat] : ~ rep_assn(P,H2) ) ).

% ent_false_iff
tff(fact_9622_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_9623_ent__false,axiom,
    ! [P: assn] : entails(bot_bot_assn,P) ).

% ent_false
tff(fact_9624_lsb__integer__code,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(least_7544222001954398261nteger,X))
    <=> pp(aa_nat_bool(bit_se9216721137139052372nteger(X),zero_zero_nat)) ) ).

% lsb_integer_code
tff(fact_9625_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd: bool,Ve: bool] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va2,Vb2),Vc)),vEBT_Leafi(Vd,Ve)) = bot_bot_assn ).

% vebt_assn_raw.simps(3)
tff(fact_9626_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: bool,Ve: bool,V: option4927543243414619207at_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf(Vd,Ve)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V,Va2,Vb2),Vc)) = bot_bot_assn ).

% vebt_assn_raw.simps(4)
tff(fact_9627_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ! [Uv: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv))) )
         => ( ! [Uu: bool] :
                ( ( X = vEBT_Leaf(Uu,fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu,fTrue))) )
           => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va3,Vb),Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9628_Bit__integer_Oabs__eq,axiom,
    ! [Xa4: int,X: bool] : bits_Bit_integer(aa_int_Code_integer(code_integer_of_int,Xa4),X) = aa_int_Code_integer(code_integer_of_int,aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,X)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Xa4))) ).

% Bit_integer.abs_eq
tff(fact_9629_vebt__assn__raw_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa4: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,X),Xa4) = Y )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(X,Xa4))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ! [Ai: bool,Bi: bool] :
                  ( ( Xa4 = vEBT_Leafi(Ai,Bi) )
                 => ( ( Y = aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_bool_bool(fequal_bool(Ai),A4)),aa_bool_bool(fequal_bool(Bi),B4))) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf(A4,B4),vEBT_Leafi(Ai,Bi))) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xa4 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                   => ( ( Y = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_bool_bool(fconj(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,Mmoi),Mmo)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Degi),Deg2)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_rd(Tree_list),Tree_array))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va,Vb3),Vc3) )
                 => ! [Vd3: bool,Ve3: bool] :
                      ( ( Xa4 = vEBT_Leafi(Vd3,Ve3) )
                     => ( ( Y = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va,Vb3),Vc3),vEBT_Leafi(Vd3,Ve3))) ) ) )
             => ~ ! [Vd3: bool,Ve3: bool] :
                    ( ( X = vEBT_Leaf(Vd3,Ve3) )
                   => ! [V2: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa4 = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va,Vb3),Vc3) )
                       => ( ( Y = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf(Vd3,Ve3),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va,Vb3),Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_9630_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
      & ( ( N != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_re,N)),X) ) ) ) ).

% root_def
tff(fact_9631_dup__1,axiom,
    code_dup(one_one_Code_integer) = numera6620942414471956472nteger(bit0(one)) ).

% dup_1
tff(fact_9632_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( pp(aa_Code_integer_bool(bits_b8758750999018896077nteger,I))
    <=> ( modulo364778990260209775nteger(I,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9633_dup_Oabs__eq,axiom,
    ! [X: int] : code_dup(aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(plus_plus_int(X),X)) ).

% dup.abs_eq
tff(fact_9634_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( pp(aa_Code_integer_bool(bits_b8758750999018896077nteger,aa_int_Code_integer(code_integer_of_int,X)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X)) ) ).

% bin_last_integer.abs_eq
tff(fact_9635_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9636_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_9637_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% push_bit_of_Suc_0
tff(fact_9638_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se545348938243370406it_int(zero_zero_nat),I) = I ).

% push_bit_int_code(1)
tff(fact_9639_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(N),K)) ).

% push_bit_nat_eq
tff(fact_9640_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),aa_int_int(bit_se545348938243370406it_int(N),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9641_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9642_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9643_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(M),K)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9644_Bit__Operations_Oset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(bit_se1409905431419307370or_int(K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9645_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q2)),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9646_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(N),K) = aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% flip_bit_int_def
tff(fact_9647_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).

% unset_bit_int_def
tff(fact_9648_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(N),K) = aa_int_int(times_times_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ).

% push_bit_int_def
tff(fact_9649_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,fTrue) = plus_p5714425477246183910nteger(bit_se7788150548672797655nteger(one_one_nat,I),one_one_Code_integer) ).

% Bit_integer_code(2)
tff(fact_9650_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) ).

% push_bit_nat_def
tff(fact_9651_push__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),N)) ).

% push_bit_minus_one
tff(fact_9652_shiftl__integer__conv__mult__pow2,axiom,
    ! [N: nat,X: code_integer] : bit_se7788150548672797655nteger(N,X) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(X),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),N)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9653_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] : bits_b2549910563261871055nteger(aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9654_upto__aux__rec,axiom,
    ! [J2: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I))
       => ( upto_aux(I,J2,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I))
       => ( upto_aux(I,J2,Js) = upto_aux(I,aa_int_int(minus_minus_int(J2),one_one_int),cons_int(J2,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_9655_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = divide6298287555418463151nteger(I,numera6620942414471956472nteger(bit0(one))) ).

% bin_rest_integer_code
tff(fact_9656_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N),K),L) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_concat_bit(N,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),L))) ).

% concat_bit_Suc
tff(fact_9657_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_9658_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_concat_bit(N,K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(N),K) ).

% concat_bit_of_zero_2
tff(fact_9659_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(N,K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_9660_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_concat_bit(N,K),L)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).

% concat_bit_negative_iff
tff(fact_9661_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa_int_int(bit_concat_bit(N,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(N),L) ).

% concat_bit_of_zero_1
tff(fact_9662_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(N,K),aa_int_int(bit_concat_bit(M,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N),aa_int_int(bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_9663_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S: int] :
      ( ( aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(bit_concat_bit(N,R2),S) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(bit_se2923211474154528505it_int(N),R2) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_9664_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B: int] : bit_concat_bit(N,aa_int_int(bit_se2923211474154528505it_int(N),B)) = bit_concat_bit(N,B) ).

% concat_bit_take_bit_eq
tff(fact_9665_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(plus_plus_int(aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).

% concat_bit_eq
tff(fact_9666_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_concat_bit(N,K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).

% concat_bit_def
tff(fact_9667_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(M,K),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9668_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(bit_concat_bit(N,K),aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9669_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,bit0(one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),size_size_list_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_9670_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M5))
         => ! [N7: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N7))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X7,M5)),aa_nat_real(X7,N7)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9671_sdiv__int__div__0,axiom,
    ! [X: int] : aa_int_int(signed6714573509424544716de_int(X),zero_zero_int) = zero_zero_int ).

% sdiv_int_div_0
tff(fact_9672_sdiv__int__0__div,axiom,
    ! [X: int] : aa_int_int(signed6714573509424544716de_int(zero_zero_int),X) = zero_zero_int ).

% sdiv_int_0_div
tff(fact_9673_int__sdiv__simps_I2_J,axiom,
    ! [A: int] : aa_int_int(signed6714573509424544716de_int(A),zero_zero_int) = zero_zero_int ).

% int_sdiv_simps(2)
tff(fact_9674_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(signed6714573509424544716de_int(A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_9675_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(signed6714573509424544716de_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% sdiv_int_numeral_numeral
tff(fact_9676_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(signed6714573509424544716de_int(A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_9677_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(signed6714573509424544716de_int(A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(signed6714573509424544716de_int(A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(times_times_int(A),B)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_9678_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(signed6714573509424544716de_int(K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_9679_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,top_top_assn)
     => entails(P,aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% entails_solve_init(1)
tff(fact_9680_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9681_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( fI_QUERY(P,Q,F2)
    <=> entails(P,aa_assn_assn(times_times_assn(Q),F2)) ) ).

% FI_QUERY_def
tff(fact_9682_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( fI_QUERY(P,Q,F2)
     => entails(P,aa_assn_assn(times_times_assn(Q),F2)) ) ).

% frame_inference_init
tff(fact_9683_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,one_one_assn)
     => entails(P,Q) ) ).

% entails_solve_init(2)
tff(fact_9684_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9685_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9686_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => pp(aa_set_int_bool(member_int(aa_int_int(signed6292675348222524329lo_int(A),B)),set_or1266510415728281911st_int(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B))),one_one_int),aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,B)),one_one_int)))) ) ).

% smod_int_range
tff(fact_9687_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq
tff(fact_9688_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D)) ) ).

% valid_eq2
tff(fact_9689_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,T),D))
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,T),D)) ) ).

% valid_eq1
tff(fact_9690_smod__int__0__mod,axiom,
    ! [X: int] : aa_int_int(signed6292675348222524329lo_int(zero_zero_int),X) = zero_zero_int ).

% smod_int_0_mod
tff(fact_9691_smod__int__mod__0,axiom,
    ! [X: int] : aa_int_int(signed6292675348222524329lo_int(X),zero_zero_int) = X ).

% smod_int_mod_0
tff(fact_9692_smod__int__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(signed6292675348222524329lo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% smod_int_numeral_numeral
tff(fact_9693_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: bool,Uv2: bool,D: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_VEBT_valid,vEBT_Leaf(Uu2,Uv2)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9694_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(signed6292675348222524329lo_int(A),B))) ) ) ).

% smod_int_compares(8)
tff(fact_9695_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A),B)),zero_zero_int)) ) ) ).

% smod_int_compares(7)
tff(fact_9696_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A),B))) ) ) ).

% smod_int_compares(6)
tff(fact_9697_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A),B)),zero_zero_int)) ) ) ).

% smod_int_compares(4)
tff(fact_9698_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A),B))) ) ) ).

% smod_int_compares(2)
tff(fact_9699_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A),B)),B)) ) ) ).

% smod_int_compares(1)
tff(fact_9700_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( aa_int_int(signed6292675348222524329lo_int(A),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% smod_mod_positive
tff(fact_9701_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(signed6292675348222524329lo_int(A),B))) ) ) ).

% smod_int_compares(3)
tff(fact_9702_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A),B)),aa_int_int(uminus_uminus_int,B))) ) ) ).

% smod_int_compares(5)
tff(fact_9703_uint32_Osize__eq__length,axiom,
    aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9704_len__num0,axiom,
    ! [Uu2: itself_Numeral_num0] : type_l4264026598287037464l_num0(Uu2) = zero_zero_nat ).

% len_num0
tff(fact_9705_len__of__finite__2__def,axiom,
    ! [X: itself_finite_2] : type_l31302759751748492nite_2(X) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% len_of_finite_2_def
tff(fact_9706_len__of__finite__3__def,axiom,
    ! [X: itself_finite_3] : type_l31302759751748493nite_3(X) = aa_num_nat(numeral_numeral_nat,bit0(bit0(one))) ).

% len_of_finite_3_def
tff(fact_9707_bij__betw__Suc,axiom,
    ! [M8: set_nat,N4: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M8),N4))
    <=> ( image_nat_nat(suc,M8) = N4 ) ) ).

% bij_betw_Suc
tff(fact_9708_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J2)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J2)) ).

% image_Suc_atLeastLessThan
tff(fact_9709_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J2)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J2)) ).

% image_Suc_atLeastAtMost
tff(fact_9710_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_9711_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).

% zero_notin_Suc_image
tff(fact_9712_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_9713_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa_set_int_bool(member_int(aa_nat_int(semiri1314217659103216013at_int,N)),image_int_int(abs_abs_int,top_top_set_int))) ).

% int_in_range_abs
tff(fact_9714_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_9715_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9716_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9717_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9718_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9719_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_9720_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rf,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9721_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_rg(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(minus_minus_int(U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9722_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_rh(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,minus_8373710615458151222nteger(U,L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9723_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9724_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9725_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,S),M))
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Listy))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_invar_vebt,X2),N)) )
       => ( ( M = aa_nat_nat(suc,N) )
         => ( ! [X2: vEBT_VEBT] :
                ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9726_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( member_VEBT_VEBT(T,set_VEBT_VEBT2(TreeList))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList)))))) ) ).

% height_compose_list
tff(fact_9727_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),lattic8265883725875713057ax_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ).

% max_ins_scaled
tff(fact_9728_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))) ) ).

% height_i_max
tff(fact_9729_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))))) ) ).

% max_idx_list
tff(fact_9730_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9731_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_rj(N),M))) ) ) ) ).

% divide_nat_def
tff(fact_9732_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9733_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X) = Y )
     => ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9734_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),X))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Y = zero_zero_nat )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf(A4,B4))) ) )
         => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2))) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9735_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_9736_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_ag,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9737_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_9738_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9739_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_9740_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus'
tff(fact_9741_min__minus,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus
tff(fact_9742_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_9743_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_9744_rev__min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)) = A ).

% rev_min_pm1
tff(fact_9745_rev__min__pm,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% rev_min_pm
tff(fact_9746_min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)) = A ).

% min_pm1
tff(fact_9747_min__pm,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% min_pm
tff(fact_9748_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_9749_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9750_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_9751_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_9752_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_9753_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] : aa_int_int(bit_concat_bit(M,aa_int_int(bit_concat_bit(N,K),L)),R2) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9754_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_concat_bit(N,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9755_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N))) ).

% mod_mod_power
tff(fact_9756_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat,Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9757_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_mi8085742599997312461d_enat(Q2,zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9758_shiftl__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N) ).

% shiftl_Suc_0
tff(fact_9759_shiftr__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_bool_nat(zero_n2687167440665602831ol_nat,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).

% shiftr_Suc_0
tff(fact_9760_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( X != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9761_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9762_inj__Suc,axiom,
    ! [N4: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N4)) ).

% inj_Suc
tff(fact_9763_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( pp(aa_set_nat_bool(member_nat(N2),N4))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,K)),N4)) ) ).

% inj_on_diff_nat
tff(fact_9764_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z5))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),B))
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F5,Z5)) ) ) ) ) ).

% MVT2
tff(fact_9765_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => ( aa_real_real(F,X) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9766_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9767_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H5))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9768_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H5)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9769_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9770_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9771_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9772_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9773_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9774_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_or1222579329274155063t_real(A,B)))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),set_or1222579329274155063t_real(A,B)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X2))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9775_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H5))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9776_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H5)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9777_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,X: real,Y: real] :
      ( ! [X2: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9778_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9779_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H5)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H5))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9780_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H5)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H5)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9781_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H5)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H5))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9782_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
            & ! [H5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H5))
               => ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H5)),S2))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H5),D3))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H5)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9783_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(bit0(one))) ) ) ) ).

% DERIV_const_average
tff(fact_9784_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9785_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,X))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9786_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9787_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rk,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_9788_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X: real,N: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rl(G),N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9789_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_rm,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9790_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_rn(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9791_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9792_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ro(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9793_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(bit0(one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9794_DERIV__arctan,axiom,
    ! [X: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9795_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_9796_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(bit0(one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),numeral_numeral_real(bit0(one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9797_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9798_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9799_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9800_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9801_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,bit0(one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9802_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rp(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9803_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rp(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9804_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9805_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T3: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_pp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9806_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ? [T3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_pp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9807_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T3: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_pp(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9808_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rp(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9809_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T3: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
       => ? [T3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rp(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9810_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ? [T3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),C))
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rq(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9811_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
             => ? [T3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),B))
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rq(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9812_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T3: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T3))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
                 => ( ( X != C )
                   => ? [T3: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T3))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T3))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T3),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rr(Diff),C),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T3)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9813_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_re,N)),top_top_set_real)) ) ).

% inj_sgn_power
tff(fact_9814_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M3: nat,T3: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T3))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T3),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T6: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_rt(N),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ru(Diff),M2),T6),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9815_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_rv,suminf_real(aa_real_fun_nat_real(aTP_Lamm_rw,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9816_DERIV__power__series_H,axiom,
    ! [R3: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( pp(aa_set_real_bool(member_real(X2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3)))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_rx(F),X2)) )
     => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
         => has_fi5821293074295781190e_real(aTP_Lamm_rz(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_rx(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9817_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_set_real_bool(member_real(X),set_or1633881224788618240n_real(A,B)))
       => ( pp(aa_set_real_bool(member_real(Y),set_or1633881224788618240n_real(A,B)))
         => ( ! [X2: real] :
                ( pp(aa_set_real_bool(member_real(X2),set_or1633881224788618240n_real(A,B)))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9818_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F5: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_sa(F),N2),aa_nat_real(aa_real_fun_nat_real(F5,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),set_or1633881224788618240n_real(A,B)))
           => summable_real(aa_real_fun_nat_real(F,X2)) )
       => ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(A,B)))
         => ( summable_real(aa_real_fun_nat_real(F5,X0))
           => ( summable_real(L5)
             => ( ! [N2: nat,X2: real,Y3: real] :
                    ( pp(aa_set_real_bool(member_real(X2),set_or1633881224788618240n_real(A,B)))
                   => ( pp(aa_set_real_bool(member_real(Y3),set_or1633881224788618240n_real(A,B)))
                     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X2),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y3))))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_sb(F),suminf_real(aa_real_fun_nat_real(F5,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9819_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9820_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [R: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X3))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9821_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R)) )
               => ( aa_real_real(F,X3) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9822_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [R: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9823_isCont__real__sqrt,axiom,
    ! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9824_isCont__real__root,axiom,
    ! [X: real,N: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),aa_nat_fun_real_real(root,N)) ).

% isCont_real_root
tff(fact_9825_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9826_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ( ! [Z5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z5),B))
               => ( aa_real_real(G,aa_real_real(F,Z5)) = Z5 ) ) )
         => ( ! [Z5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z5),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z5,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9827_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_9828_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_9829_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9830_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_sc,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9831_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
           => ( ! [Y3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B))
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9832_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9833_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9834_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X2: real] :
            ( pp(aa_set_real_bool(member_real(X2),set_or1633881224788618240n_real(B,X)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).

% LIM_less_bound
tff(fact_9835_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9836_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z5),X))),D))
           => ( aa_real_real(G,aa_real_real(F,Z5)) = Z5 ) )
       => ( ! [Z5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z5),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z5,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9837_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [Z5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z5))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z5),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z5,top_top_set_real),F) ) )
       => ( ! [Z5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z5))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z5),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z5,top_top_set_real),G) ) )
         => ( ! [Z5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z5),topolo2177554685111907308n_real(Z5,top_top_set_real)) ) )
           => ( ! [Z5: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z5))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z5),topolo2177554685111907308n_real(Z5,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F5,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9838_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N9: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_sd(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N9)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N9)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9839_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
         => ! [N9: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_sd(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N9))),groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N9)),one_one_nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9840_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9841_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_av,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9842_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_se,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9843_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sf(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N9)),L3))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G,N9)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9844_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R: real] :
        ? [N11: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N11),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R),aa_nat_real(X7,N2))) )
     => filterlim_nat_real(aTP_Lamm_sg(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9845_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_sh,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9846_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_si,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9847_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sj,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9848_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sk,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9849_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),L))
       => ( ! [E2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
             => ? [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N9)),E2))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9850_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9851_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sl(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9852_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9853_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9854_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9855_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9856_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_so,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9857_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sp,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9858_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_sd(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9859_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => summable_real(aTP_Lamm_sd(A)) ) ) ) ).

% summable
tff(fact_9860_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sq(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_sr(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9861_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_ss(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_sr(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9862_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_st(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sd(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9863_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ef,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9864_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)))),suminf_real(aTP_Lamm_sd(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9865_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_st(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sd(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9866_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N9)))),L3))
              & filterlim_nat_real(aTP_Lamm_st(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N9: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N9)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_su(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9867_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_su(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sd(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9868_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_sd(A))),groups6591440286371151544t_real(aTP_Lamm_sd(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9869_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
       => ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
         => filterlim_nat_real(aTP_Lamm_su(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sd(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9870_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C4: real] :
        ! [X5: real] : aa_real_real(F,X5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X5),C4) ) ).

% real_bounded_linear
tff(fact_9871_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9872_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sv,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9873_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_9874_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
tff(fact_9875_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))))) ).

% filterlim_tan_at_right
tff(fact_9876_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_9877_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
tff(fact_9878_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => pp(aa_set_real_bool(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real))) ) ).

% log_inj
tff(fact_9879_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
tff(fact_9880_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9881_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: fun_real_real,F2: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sw(N),F),at_bot_real,F2) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9882_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: fun_real_real,F2: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),N))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sw(N),F),at_top_real,F2) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9883_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9884_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_9885_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9886_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_sx,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9887_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
tff(fact_9888_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
tff(fact_9889_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_sy,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9890_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sz,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_9891_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))) ).

% filterlim_tan_at_left
tff(fact_9892_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_9893_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X2))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9894_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9895_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_td(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9896_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_te(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9897_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_te(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9898_eventually__at__right__to__0,axiom,
    ! [P: fun_real_bool,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tf(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9899_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tg(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9900_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tg(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9901_eventually__at__right__to__top,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_th(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_9902_eventually__at__top__to__right,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_th(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_9903_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F5),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_9904_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F2: 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(aTP_Lamm_ta(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),F2,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),F2,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9905_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F5),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_9906_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F5),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_9907_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F5),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_9908_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9909_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_ta(G2),at_top_real)
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),at_top_real)
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9910_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F2: 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(aTP_Lamm_ta(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),F2,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),F2,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9911_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F5: fun_real_real,F2: 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(aTP_Lamm_ta(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F0),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),F2,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F0),G0),F2,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9912_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F2: 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(aTP_Lamm_ta(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),F2,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),F2,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9913_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F5),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_9914_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F5),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_9915_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9916_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_ta(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(F),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G2),F5),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9917_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9918_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9919_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(inverse_inverse_rat(P3)) = produc4245557441103728435nt_int(aTP_Lamm_tj,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9920_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% quotient_of_number(3)
tff(fact_9921_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_9922_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(5)
tff(fact_9923_quotient__of__div,axiom,
    ! [R2: rat,N: int,D: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(N,D) )
     => ( R2 = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,N),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9924_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : divide_divide_rat(Q2,R2) = aa_rat_rat(times_times_rat(Q2),inverse_inverse_rat(R2)) ).

% divide_rat_def
tff(fact_9925_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9926_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q2: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% quotient_of_denom_pos
tff(fact_9927_rat__floor__code,axiom,
    ! [P3: rat] : archim3151403230148437115or_rat(P3) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P3)) ).

% rat_floor_code
tff(fact_9928_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_9929_rat__less__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q2))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_tl(Q2)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_9930_rat__plus__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(plus_plus_rat(P3,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_tn(Q2),quotient_of(P3)) ).

% rat_plus_code
tff(fact_9931_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X2: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G) )
           => ( ! [X2: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X2,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9932_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(product_Pair_int_int(P3,zero_zero_int)) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% normalize_denom_zero
tff(fact_9933_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q2),zero_zero_int))
     => ( normalize(product_Pair_int_int(P3,Q2)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9934_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : minus_minus_rat(Q2,R2) = plus_plus_rat(Q2,uminus_uminus_rat(R2)) ).

% diff_rat_def
tff(fact_9935_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S3))
           => ! [T3: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T3))
               => ( R2 != plus_plus_rat(S3,T3) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9936_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q2: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).

% normalize_denom_pos
tff(fact_9937_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P3: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P3,Q2)) = normalize(product_Pair_int_int(R2,S)) )
         => ( aa_int_int(times_times_int(P3),S) = aa_int_int(times_times_int(R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9938_rat__divide__code,axiom,
    ! [P3: rat,Q2: rat] : quotient_of(divide_divide_rat(P3,Q2)) = produc4245557441103728435nt_int(aTP_Lamm_tp(Q2),quotient_of(P3)) ).

% rat_divide_code
tff(fact_9939_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_9940_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ? [L3: real,Z5: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),B))
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z5,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_9941_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),B))
                & ! [X3: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z5),X3) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9942_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ~ ! [Xi3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Xi3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi3),B))
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9943_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(product_Pair_int_int(A,zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_9944_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(product_Pair_int_int(zero_zero_int,A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_9945_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9946_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9947_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9948_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9949_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X2))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),B))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
           => ? [Z5: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z5))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z5),B))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z5,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9950_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_9951_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_9952_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J2))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9953_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9954_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_9955_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J2))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J2)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9956_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J2))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J2)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9957_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9958_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J2))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9959_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(uminus1351360451143612070nteger(K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_tq,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9960_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one)))) ).

% csqrt.simps(1)
tff(fact_9961_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9962_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(numera6620942414471956472nteger(K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9963_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa4: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(X,Xa4)) = aa_int_int(plus_plus_int(code_int_of_integer(X)),code_int_of_integer(Xa4)) ).

% plus_integer.rep_eq
tff(fact_9964_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = numeral_numeral_real(V) ).

% complex_Re_numeral
tff(fact_9965_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa4: code_integer] : code_int_of_integer(divide6298287555418463151nteger(X,Xa4)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(X)),code_int_of_integer(Xa4)) ).

% divide_integer.rep_eq
tff(fact_9966_Re__divide__of__nat,axiom,
    ! [Z: complex,N: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% Re_divide_of_nat
tff(fact_9967_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),R2) ).

% Re_divide_of_real
tff(fact_9968_Re__sgn,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Re_sgn
tff(fact_9969_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),numeral_numeral_real(W)) ).

% Re_divide_numeral
tff(fact_9970_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( aa_complex_real(re,Y) = zero_zero_real )
       => ( aa_real_real(cos_real,aa_complex_real(arg,Y)) = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_9971_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R2),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,X)) ).

% scaleR_complex.simps(1)
tff(fact_9972_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y)) ).

% plus_complex.simps(1)
tff(fact_9973_dup_Orep__eq,axiom,
    ! [X: code_integer] : code_int_of_integer(code_dup(X)) = aa_int_int(plus_plus_int(code_int_of_integer(X)),code_int_of_integer(X)) ).

% dup.rep_eq
tff(fact_9974_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_9975_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).

% int_of_integer_less_iff
tff(fact_9976_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa4: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa4))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa4))) ) ).

% less_integer.rep_eq
tff(fact_9977_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_9978_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_9979_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa4: code_integer] : code_int_of_integer(bit_se8260200283734997820nteger(X,Xa4)) = aa_int_int(bit_se4203085406695923979it_int(X),code_int_of_integer(Xa4)) ).

% unset_bit_integer.rep_eq
tff(fact_9980_Re__csqrt,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))) ).

% Re_csqrt
tff(fact_9981_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real))
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9982_bin__last__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(bits_b8758750999018896077nteger,X))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),code_int_of_integer(X))) ) ).

% bin_last_integer.rep_eq
tff(fact_9983_bin__rest__integer_Orep__eq,axiom,
    ! [X: code_integer] : code_int_of_integer(bits_b2549910563261871055nteger(X)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(X)),aa_num_int(numeral_numeral_int,bit0(one))) ).

% bin_rest_integer.rep_eq
tff(fact_9984_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).

% cos_n_Re_cis_pow_n
tff(fact_9985_Bit__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa4: bool] : code_int_of_integer(bits_Bit_integer(X,Xa4)) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,Xa4)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),code_int_of_integer(X))) ).

% Bit_integer.rep_eq
tff(fact_9986_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).

% divmod_integer_def
tff(fact_9987_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = one ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = produc7336495610019696514er_num(aTP_Lamm_tr,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one)))) ) ) ) ).

% num_of_integer_code
tff(fact_9988_csqrt_Ocode,axiom,
    ! [Z: complex] : aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one)))))) ).

% csqrt.code
tff(fact_9989_complex__Im__fact,axiom,
    ! [N: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,N)) = zero_zero_real ).

% complex_Im_fact
tff(fact_9990_complex__Im__of__int,axiom,
    ! [Z: int] : aa_complex_real(im,aa_int_complex(ring_17405671764205052669omplex,Z)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_9991_Im__complex__of__real,axiom,
    ! [Z: real] : aa_complex_real(im,aa_real_complex(real_V4546457046886955230omplex,Z)) = zero_zero_real ).

% Im_complex_of_real
tff(fact_9992_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(X),N)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_9993_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9994_complex__Im__of__nat,axiom,
    ! [N: nat] : aa_complex_real(im,aa_nat_complex(semiri8010041392384452111omplex,N)) = zero_zero_real ).

% complex_Im_of_nat
tff(fact_9995_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),R2) ).

% Im_divide_of_real
tff(fact_9996_Im__sgn,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Im_sgn
tff(fact_9997_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),N) ) ) ).

% Re_power_real
tff(fact_9998_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),numeral_numeral_real(W)) ).

% Im_divide_numeral
tff(fact_9999_Im__divide__of__nat,axiom,
    ! [Z: complex,N: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% Im_divide_of_nat
tff(fact_10000_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X)))
       => ( aa_complex_complex(csqrt,X) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_10001_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,X)),zero_zero_real))
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).

% csqrt_minus
tff(fact_10002_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),zero_zero_real))
       => ( aa_complex_complex(csqrt,X) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_10003_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( member_complex(Z,ring_1_Ints_complex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I2: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I2) ) ) ).

% complex_is_Int_iff
tff(fact_10004_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_10005_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R2),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,X)) ).

% scaleR_complex.simps(2)
tff(fact_10006_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_10007_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y)) ).

% plus_complex.simps(2)
tff(fact_10008_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_10009_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(im,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(re,Z)) ) ) ).

% cmod_eq_Re
tff(fact_10010_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(re,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(im,Z)) ) ) ).

% cmod_eq_Im
tff(fact_10011_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_real_real(abs_abs_real,aa_complex_real(re,Z)) = aa_complex_real(real_V1022390504157884413omplex,Z) )
     => ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% Im_eq_0
tff(fact_10012_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_10013_plus__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(plus_plus_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).

% plus_complex.code
tff(fact_10014_scaleR__complex_Ocode,axiom,
    ! [R2: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(R2),X) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,X))) ).

% scaleR_complex.code
tff(fact_10015_csqrt__principal,axiom,
    ! [Z: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z)))) ) ) ).

% csqrt_principal
tff(fact_10016_cmod__le,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z))))) ).

% cmod_le
tff(fact_10017_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).

% sin_n_Im_cis_pow_n
tff(fact_10018_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_10019_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_10020_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_10021_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_10022_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% cmod_power2
tff(fact_10023_Im__power2,axiom,
    ! [X: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).

% Im_power2
tff(fact_10024_Re__power2,axiom,
    ! [X: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% Re_power2
tff(fact_10025_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_10026_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% norm_complex_def
tff(fact_10027_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% inverse_complex.simps(1)
tff(fact_10028_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ) ).

% complex_neq_0
tff(fact_10029_Re__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% Re_divide
tff(fact_10030_csqrt__square,axiom,
    ! [B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,B)))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B))) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,bit0(one)))) = B ) ) ).

% csqrt_square
tff(fact_10031_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,bit0(one))) = Z )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,W)))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W))) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_10032_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% inverse_complex.simps(2)
tff(fact_10033_Im__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% Im_divide
tff(fact_10034_complex__abs__le__norm,axiom,
    ! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ).

% complex_abs_le_norm
tff(fact_10035_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,bit0(one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_10036_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa_complex_complex(invers8013647133539491842omplex,X) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% inverse_complex.code
tff(fact_10037_Complex__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,bit0(one)))))) ).

% Complex_divide
tff(fact_10038_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one))))) ).

% csqrt.simps(2)
tff(fact_10039_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% Im_Reals_divide
tff(fact_10040_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_ts,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_10041_nat__of__integer__numeral,axiom,
    ! [N: num] : code_nat_of_integer(numera6620942414471956472nteger(N)) = aa_num_nat(numeral_numeral_nat,N) ).

% nat_of_integer_numeral
tff(fact_10042_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_10043_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_10044_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(re,R2)) ) ) ).

% Re_divide_Reals
tff(fact_10045_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(re,R2)) ) ) ).

% Im_divide_Reals
tff(fact_10046_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( member_complex(Z,real_V2521375963428798218omplex)
    <=> ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_10047_Complex__in__Reals,axiom,
    ! [X: real] : member_complex(aa_real_complex(complex2(X),zero_zero_real),real_V2521375963428798218omplex) ).

% Complex_in_Reals
tff(fact_10048_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_10049_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,code_nat_of_integer(X)),code_nat_of_integer(Y)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_10050_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))) ) ) ).

% Re_Reals_divide
tff(fact_10051_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% complex_mult_cnj
tff(fact_10052_card__Collect__less__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dw,N))) = N ).

% card_Collect_less_nat
tff(fact_10053_card__atMost,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_10054_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_10055_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_divide
tff(fact_10056_card__Collect__le__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tt,N))) = aa_nat_nat(suc,N) ).

% card_Collect_le_nat
tff(fact_10057_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_add
tff(fact_10058_card__UNIV__bool,axiom,
    finite_card_bool(top_top_set_bool) = aa_num_nat(numeral_numeral_nat,bit0(one)) ).

% card_UNIV_bool
tff(fact_10059_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_10060_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_10061_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_10062_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_10063_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),aa_int_int(plus_plus_int(L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_10064_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_10065_card__less__Suc2,axiom,
    ! [M8: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tu(M8),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tv(M8),I))) ) ) ).

% card_less_Suc2
tff(fact_10066_card__less__Suc,axiom,
    ! [M8: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tu(M8),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tv(M8),I))) ) ) ).

% card_less_Suc
tff(fact_10067_card__less,axiom,
    ! [M8: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M8))
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_tv(M8),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_10068_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2)))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_10069_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_10070_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_10071_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(N4)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_10072_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(S2)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S2)))) ) ).

% card_le_Suc_Max
tff(fact_10073_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),S2))) ).

% card_sum_le_nat_sum
tff(fact_10074_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qo(C),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_10075_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( finite_card_complex(collect_complex(aTP_Lamm_dv(N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_10076_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Re_complex_div_lt_0
tff(fact_10077_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Re_complex_div_gt_0
tff(fact_10078_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Re_complex_div_ge_0
tff(fact_10079_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Re_complex_div_le_0
tff(fact_10080_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Im_complex_div_lt_0
tff(fact_10081_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Im_complex_div_gt_0
tff(fact_10082_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Im_complex_div_ge_0
tff(fact_10083_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Im_complex_div_le_0
tff(fact_10084_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% complex_mod_mult_cnj
tff(fact_10085_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) )
      & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ) ).

% complex_div_gt_0
tff(fact_10086_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,bit0(one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_10087_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_10088_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_10089_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% complex_div_cnj
tff(fact_10090_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_10091_card__num0,axiom,
    finite6454714172617411596l_num0(top_to3689904424835650196l_num0) = zero_zero_nat ).

% card_num0
tff(fact_10092_card__nat,axiom,
    finite_card_nat(top_top_set_nat) = zero_zero_nat ).

% card_nat
tff(fact_10093_card__literal,axiom,
    finite_card_literal(top_top_set_literal) = zero_zero_nat ).

% card_literal
tff(fact_10094_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))) ).

% card_UNIV_char
tff(fact_10095_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% UNIV_char_of_nat
tff(fact_10096_inj__on__char__of__nat,axiom,
    pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_10097_char_Osize_I2_J,axiom,
    ! [X1: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_size_char(char2(X1,X22,X32,X42,X52,X62,X72,X8)) = zero_zero_nat ).

% char.size(2)
tff(fact_10098_nat__of__char__less__256,axiom,
    ! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% nat_of_char_less_256
tff(fact_10099_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% range_nat_of_char
tff(fact_10100_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(B72)),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B62))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B52))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B42))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B32))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B22))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B1))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger(B0)) ).

% integer_of_char_code
tff(fact_10101_char_Osize__gen,axiom,
    ! [X1: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_char(char2(X1,X22,X32,X42,X52,X62,X72,X8)) = zero_zero_nat ).

% char.size_gen
tff(fact_10102_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_10103_less__eq__char__simp,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool,C0: bool,C1: bool,C22: bool,C32: bool,C42: bool,C52: bool,C6: bool,C7: bool] :
      ( ord_less_eq_char(char2(B0,B1,B22,B32,B42,B52,B62,B72),char2(C0,C1,C22,C32,C42,C52,C6,C7))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(foldr_bool_nat(aTP_Lamm_tw,cons_bool(B0,cons_bool(B1,cons_bool(B22,cons_bool(B32,cons_bool(B42,cons_bool(B52,cons_bool(B62,cons_bool(B72,nil_bool))))))))),zero_zero_nat)),aa_nat_nat(foldr_bool_nat(aTP_Lamm_tw,cons_bool(C0,cons_bool(C1,cons_bool(C22,cons_bool(C32,cons_bool(C42,cons_bool(C52,cons_bool(C6,cons_bool(C7,nil_bool))))))))),zero_zero_nat))) ) ).

% less_eq_char_simp
tff(fact_10104_less__char__simp,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool,C0: bool,C1: bool,C22: bool,C32: bool,C42: bool,C52: bool,C6: bool,C7: bool] :
      ( ord_less_char(char2(B0,B1,B22,B32,B42,B52,B62,B72),char2(C0,C1,C22,C32,C42,C52,C6,C7))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(foldr_bool_nat(aTP_Lamm_tw,cons_bool(B0,cons_bool(B1,cons_bool(B22,cons_bool(B32,cons_bool(B42,cons_bool(B52,cons_bool(B62,cons_bool(B72,nil_bool))))))))),zero_zero_nat)),aa_nat_nat(foldr_bool_nat(aTP_Lamm_tw,cons_bool(C0,cons_bool(C1,cons_bool(C22,cons_bool(C32,cons_bool(C42,cons_bool(C52,cons_bool(C6,cons_bool(C7,nil_bool))))))))),zero_zero_nat))) ) ).

% less_char_simp
tff(fact_10105_less__char__def,axiom,
    ! [C1: char,C22: char] :
      ( ord_less_char(C1,C22)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22))) ) ).

% less_char_def
tff(fact_10106_upto_Opsimps,axiom,
    ! [I: int,J2: int] :
      ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I,J2)))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
         => ( upto(I,J2) = cons_int(I,upto(aa_int_int(plus_plus_int(I),one_one_int),J2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
         => ( upto(I,J2) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_10107_upto__Nil,axiom,
    ! [I: int,J2: int] :
      ( ( upto(I,J2) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I)) ) ).

% upto_Nil
tff(fact_10108_upto__Nil2,axiom,
    ! [I: int,J2: int] :
      ( ( nil_int = upto(I,J2) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I)) ) ).

% upto_Nil2
tff(fact_10109_upto__empty,axiom,
    ! [J2: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J2),I))
     => ( upto(I,J2) = nil_int ) ) ).

% upto_empty
tff(fact_10110_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_10111_nth__upto,axiom,
    ! [I: int,K: nat,J2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(plus_plus_int(I),aa_nat_int(semiri1314217659103216013at_int,K))),J2))
     => ( aa_nat_int(nth_int(upto(I,J2)),K) = aa_int_int(plus_plus_int(I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_10112_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_10113_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_10114_length__upto,axiom,
    ! [I: int,J2: int] : size_size_list_int(upto(I,J2)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(J2),I)),one_one_int)) ).

% length_upto
tff(fact_10115_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_10116_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_10117_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_10118_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
       => ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_10119_upto__rec2,axiom,
    ! [I: int,J2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
     => ( upto(I,J2) = append_int(upto(I,aa_int_int(minus_minus_int(J2),one_one_int)),cons_int(J2,nil_int)) ) ) ).

% upto_rec2
tff(fact_10120_upto__code,axiom,
    ! [I: int,J2: int] : upto(I,J2) = upto_aux(I,J2,nil_int) ).

% upto_code
tff(fact_10121_upto__split3,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J2),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(minus_minus_int(J2),one_one_int)),cons_int(J2,upto(aa_int_int(plus_plus_int(J2),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_10122_upto__split2,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J2),K))
       => ( upto(I,K) = append_int(upto(I,J2),upto(aa_int_int(plus_plus_int(J2),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_10123_upto__split1,axiom,
    ! [I: int,J2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J2),K))
       => ( upto(I,K) = append_int(upto(I,aa_int_int(minus_minus_int(J2),one_one_int)),upto(J2,K)) ) ) ) ).

% upto_split1
tff(fact_10124_distinct__upto,axiom,
    ! [I: int,J2: int] : pp(aa_list_int_bool(distinct_int,upto(I,J2))) ).

% distinct_upto
tff(fact_10125_atLeastAtMost__upto,axiom,
    ! [I: int,J2: int] : set_or1266510415728281911st_int(I,J2) = set_int2(upto(I,J2)) ).

% atLeastAtMost_upto
tff(fact_10126_upto__aux__def,axiom,
    ! [I: int,J2: int,Js: list_int] : upto_aux(I,J2,Js) = append_int(upto(I,J2),Js) ).

% upto_aux_def
tff(fact_10127_atLeastLessThan__upto,axiom,
    ! [I: int,J2: int] : set_or4662586982721622107an_int(I,J2) = set_int2(upto(I,aa_int_int(minus_minus_int(J2),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_10128_greaterThanAtMost__upto,axiom,
    ! [I: int,J2: int] : set_or6656581121297822940st_int(I,J2) = set_int2(upto(aa_int_int(plus_plus_int(I),one_one_int),J2)) ).

% greaterThanAtMost_upto
tff(fact_10129_upto__rec1,axiom,
    ! [I: int,J2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
     => ( upto(I,J2) = cons_int(I,upto(aa_int_int(plus_plus_int(I),one_one_int),J2)) ) ) ).

% upto_rec1
tff(fact_10130_upto_Oelims,axiom,
    ! [X: int,Xa4: int,Y: list_int] :
      ( ( upto(X,Xa4) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa4))
         => ( Y = cons_int(X,upto(aa_int_int(plus_plus_int(X),one_one_int),Xa4)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa4))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_10131_upto_Osimps,axiom,
    ! [I: int,J2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
       => ( upto(I,J2) = cons_int(I,upto(aa_int_int(plus_plus_int(I),one_one_int),J2)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J2))
       => ( upto(I,J2) = nil_int ) ) ) ).

% upto.simps
tff(fact_10132_greaterThanLessThan__upto,axiom,
    ! [I: int,J2: int] : set_or5832277885323065728an_int(I,J2) = set_int2(upto(aa_int_int(plus_plus_int(I),one_one_int),aa_int_int(minus_minus_int(J2),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_10133_upto_Opelims,axiom,
    ! [X: int,Xa4: int,Y: list_int] :
      ( ( upto(X,Xa4) = Y )
     => ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa4)))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa4))
               => ( Y = cons_int(X,upto(aa_int_int(plus_plus_int(X),one_one_int),Xa4)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa4))
               => ( Y = nil_int ) ) )
           => ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa4))) ) ) ) ).

% upto.pelims
tff(fact_10134_integer__of__num_I3_J,axiom,
    ! [N: num] : code_integer_of_num(aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(code_integer_of_num(N),code_integer_of_num(N)),one_one_Code_integer) ).

% integer_of_num(3)
tff(fact_10135_sort__upto,axiom,
    ! [I: int,J2: int] : linord1735203802627413978nt_int(aTP_Lamm_as,upto(I,J2)) = upto(I,J2) ).

% sort_upto
tff(fact_10136_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_10137_integer__of__num_I2_J,axiom,
    ! [N: num] : code_integer_of_num(bit0(N)) = plus_p5714425477246183910nteger(code_integer_of_num(N),code_integer_of_num(N)) ).

% integer_of_num(2)
tff(fact_10138_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(bit0(one)) = numera6620942414471956472nteger(bit0(one)) ).

% integer_of_num_triv(2)
tff(fact_10139_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_ar,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_10140_tl__upt,axiom,
    ! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).

% tl_upt
tff(fact_10141_upt__0__eq__Nil__conv,axiom,
    ! [J2: nat] :
      ( ( upt(zero_zero_nat,J2) = nil_nat )
    <=> ( J2 = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_10142_upt__conv__Nil,axiom,
    ! [J2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I))
     => ( upt(I,J2) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_10143_length__upt,axiom,
    ! [I: nat,J2: nat] : size_size_list_nat(upt(I,J2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I) ).

% length_upt
tff(fact_10144_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_10145_upt__eq__Nil__conv,axiom,
    ! [I: nat,J2: nat] :
      ( ( upt(I,J2) = nil_nat )
    <=> ( ( J2 = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J2),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_10146_nth__upt,axiom,
    ! [I: nat,K: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2))
     => ( aa_nat_nat(nth_nat(upt(I,J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_10147_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_10148_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),U2))
     => ( ! [I3: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),I3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),U2)) )
           => ~ pp(aa_nat_bool(P,I3)) )
       => ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U2)) ) ) ) ).

% upt_filter_extend
tff(fact_10149_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(bit_se2923211474154528505it_int(N),L) )
     => ( map_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K),upt(zero_zero_nat,N)) = map_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),upt(zero_zero_nat,N)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_10150_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_tx,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_10151_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_10152_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ty,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).

% map_add_upt'
tff(fact_10153_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ty,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_10154_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_10155_upt__add__eq__append,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)) = append_nat(upt(I,J2),upt(J2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K))) ) ) ).

% upt_add_eq_append
tff(fact_10156_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_10157_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_10158_distinct__upt,axiom,
    ! [I: nat,J2: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I,J2))) ).

% distinct_upt
tff(fact_10159_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_10160_atLeastLessThan__upt,axiom,
    ! [I: nat,J2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J2) = set_nat2(upt(I,J2)) ).

% atLeastLessThan_upt
tff(fact_10161_upt__append,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J2)) = upt(zero_zero_nat,J2) ) ) ).

% upt_append
tff(fact_10162_upt__conv__Cons,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
     => ( upt(I,J2) = cons_nat(I,upt(aa_nat_nat(suc,I),J2)) ) ) ).

% upt_conv_Cons
tff(fact_10163_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q2) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_10164_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_10165_upt__eq__Cons__conv,axiom,
    ! [I: nat,J2: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J2) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J2) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_10166_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_10167_upt__rec,axiom,
    ! [I: nat,J2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
       => ( upt(I,J2) = cons_nat(I,upt(aa_nat_nat(suc,I),J2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
       => ( upt(I,J2) = nil_nat ) ) ) ).

% upt_rec
tff(fact_10168_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H) = append_nat(Is1,cons_nat(I,Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),H)) ) ) ).

% upt_eq_lel_conv
tff(fact_10169_upt__Suc__append,axiom,
    ! [I: nat,J2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
     => ( upt(I,aa_nat_nat(suc,J2)) = append_nat(upt(I,J2),cons_nat(J2,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_10170_upt__Suc,axiom,
    ! [I: nat,J2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
       => ( upt(I,aa_nat_nat(suc,J2)) = append_nat(upt(I,J2),cons_nat(J2,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J2))
       => ( upt(I,aa_nat_nat(suc,J2)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_10171_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_tz) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_10172_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_10173_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_10174_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_10175_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),bit0(M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ua,bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_10176_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_10177_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),bit0(M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ua,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M)) ).

% take_bit_num_simps(6)
tff(fact_10178_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),M))) ).

% take_bit_num_simps(7)
tff(fact_10179_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_ub(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_10180_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_10181_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_10182_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(4)
tff(fact_10183_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(8)
tff(fact_10184_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_10185_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one,aa_num_num(bit1,N)) = none_num ).

% and_not_num.simps(3)
tff(fact_10186_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one,bit0(N)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_10187_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(bit0(M),one) = some_num(bit0(M)) ).

% and_not_num.simps(4)
tff(fact_10188_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(bit0(M)) ).

% and_not_num.simps(7)
tff(fact_10189_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( bit_and_not_num(M,N) = some_num(Q2) )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_10190_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),bit0(N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_uc,bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_10191_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_10192_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_10193_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_10194_bin__last__set__bits,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),bit_bi6516823479961619367ts_int(F)))
      <=> pp(aa_nat_bool(F,zero_zero_nat)) ) ) ).

% bin_last_set_bits
tff(fact_10195_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_ud(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_10196_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(aTP_Lamm_td(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_10197_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one) = case_nat_option_num(none_num,aTP_Lamm_ue,N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_10198_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,bit0(M)) = case_nat_option_num(none_num,aTP_Lamm_uf(M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_10199_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_uj,product_Pair_nat_num(N,M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10200_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = aa_int_int(plus_plus_int(aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(F,zero_zero_nat))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),bit_bi6516823479961619367ts_int(comp_nat_bool_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_10201_bin__rest__set__bits,axiom,
    ! [F: fun_nat_bool] :
      ( bit_wf_set_bits_int(F)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_bi6516823479961619367ts_int(F)),aa_num_int(numeral_numeral_int,bit0(one))) = bit_bi6516823479961619367ts_int(comp_nat_bool_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_10202_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_tz),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_10203_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_uk),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_10204_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_10205_ATP_Olambda__1,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_eh,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu2)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10206_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_py,Uu2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_2
tff(fact_10207_ATP_Olambda__3,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_ra,Uu2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu2),numeral_numeral_real(bit0(one))))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10208_ATP_Olambda__4,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_pl,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu2)) ).

% ATP.lambda_4
tff(fact_10209_ATP_Olambda__5,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_sc,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).

% ATP.lambda_5
tff(fact_10210_ATP_Olambda__6,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_sx,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu2)),Uu2) ).

% ATP.lambda_6
tff(fact_10211_ATP_Olambda__7,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_se,Uu2) = aa_real_real(aa_nat_fun_real_real(root,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_7
tff(fact_10212_ATP_Olambda__8,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_tx,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_8
tff(fact_10213_ATP_Olambda__9,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_sh,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_9
tff(fact_10214_ATP_Olambda__10,axiom,
    ! [Uu2: int] :
      ( pp(aa_int_bool(aTP_Lamm_rc,Uu2))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Uu2)) ) ).

% ATP.lambda_10
tff(fact_10215_ATP_Olambda__11,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_rv,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_eg,Uu2)) ).

% ATP.lambda_11
tff(fact_10216_ATP_Olambda__12,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_si,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).

% ATP.lambda_12
tff(fact_10217_ATP_Olambda__13,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_uc,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).

% ATP.lambda_13
tff(fact_10218_ATP_Olambda__14,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_ua,Uu2) = some_num(bit0(Uu2)) ).

% ATP.lambda_14
tff(fact_10219_ATP_Olambda__15,axiom,
    ! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_qs,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_15
tff(fact_10220_ATP_Olambda__16,axiom,
    ! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_ue,Uu2) = some_num(one) ).

% ATP.lambda_16
tff(fact_10221_ATP_Olambda__17,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ui(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_ug(Uua),aTP_Lamm_uh(Uua),Uu2) ).

% ATP.lambda_17
tff(fact_10222_ATP_Olambda__18,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pz(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),zero_zero_real),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_18
tff(fact_10223_ATP_Olambda__19,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_tq,Uu2),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),code_int_of_integer(Uu2))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),code_int_of_integer(Uu2))),one_one_int)) ).

% ATP.lambda_19
tff(fact_10224_ATP_Olambda__20,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_tr,Uu2),Uua) = aa_num_num(aa_num_fun_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu2)),code_num_of_integer(Uu2))),aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu2)),code_num_of_integer(Uu2))),one)) ).

% ATP.lambda_20
tff(fact_10225_ATP_Olambda__21,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_ts,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu2)),code_nat_of_integer(Uu2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu2)),code_nat_of_integer(Uu2))),one_one_nat)) ).

% ATP.lambda_21
tff(fact_10226_ATP_Olambda__22,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_tj,Uu2),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu2),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,Uu2)),Uua),aa_int_int(abs_abs_int,Uu2))) ).

% ATP.lambda_22
tff(fact_10227_ATP_Olambda__23,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_uj,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_ui(Uua),Uu2) ).

% ATP.lambda_23
tff(fact_10228_ATP_Olambda__24,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ug(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ua,bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_24
tff(fact_10229_ATP_Olambda__25,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_uf(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ua,bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_25
tff(fact_10230_ATP_Olambda__26,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_co,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_26
tff(fact_10231_ATP_Olambda__27,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_px,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_27
tff(fact_10232_ATP_Olambda__28,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qb,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))) ).

% ATP.lambda_28
tff(fact_10233_ATP_Olambda__29,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_qx,Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
        & ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_29
tff(fact_10234_ATP_Olambda__30,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_qw,Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))
        & ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_30
tff(fact_10235_ATP_Olambda__31,axiom,
    ! [Uu2: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_rb(Uu2),Uua))
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu2) = aa_real_complex(cis,Uua) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).

% ATP.lambda_31
tff(fact_10236_ATP_Olambda__32,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eg,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))) ).

% ATP.lambda_32
tff(fact_10237_ATP_Olambda__33,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rw,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_33
tff(fact_10238_ATP_Olambda__34,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sd(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_34
tff(fact_10239_ATP_Olambda__35,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ql,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),aa_num_nat(numeral_numeral_nat,bit0(one))) ).

% ATP.lambda_35
tff(fact_10240_ATP_Olambda__36,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pw,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_36
tff(fact_10241_ATP_Olambda__37,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_sy,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_37
tff(fact_10242_ATP_Olambda__38,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qe,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uu2) ).

% ATP.lambda_38
tff(fact_10243_ATP_Olambda__39,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qd,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_39
tff(fact_10244_ATP_Olambda__40,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_dv(Uu2),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).

% ATP.lambda_40
tff(fact_10245_ATP_Olambda__41,axiom,
    ! [Uu2: nat,Uua: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_du,Uu2),Uua))
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) = one_one_real ) ) ).

% ATP.lambda_41
tff(fact_10246_ATP_Olambda__42,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_so,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_42
tff(fact_10247_ATP_Olambda__43,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sv,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_43
tff(fact_10248_ATP_Olambda__44,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sz,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),Uua))),Uua) ).

% ATP.lambda_44
tff(fact_10249_ATP_Olambda__45,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ef,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))),one_one_nat))) ).

% ATP.lambda_45
tff(fact_10250_ATP_Olambda__46,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_qz,Uu2),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_46
tff(fact_10251_ATP_Olambda__47,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mz(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_real(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_47
tff(fact_10252_ATP_Olambda__48,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mw(Uu2),Uua) = plus_plus_rat(aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),aa_nat_rat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_48
tff(fact_10253_ATP_Olambda__49,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_my(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_nat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_49
tff(fact_10254_ATP_Olambda__50,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mx(Uu2),Uua) = aa_int_int(plus_plus_int(aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))),aa_nat_int(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ).

% ATP.lambda_50
tff(fact_10255_ATP_Olambda__51,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lv(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,aa_nat_nat(suc,Uua))),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_51
tff(fact_10256_ATP_Olambda__52,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lt(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_52
tff(fact_10257_ATP_Olambda__53,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lu(Uu2),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu2,aa_nat_nat(suc,Uua))),aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_53
tff(fact_10258_ATP_Olambda__54,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ml(Uu2),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu2,Uua),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ).

% ATP.lambda_54
tff(fact_10259_ATP_Olambda__55,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mi(Uu2),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu2,Uua)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uua)) ).

% ATP.lambda_55
tff(fact_10260_ATP_Olambda__56,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_cp(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_56
tff(fact_10261_ATP_Olambda__57,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cq(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_57
tff(fact_10262_ATP_Olambda__58,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jc(Uu2),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_58
tff(fact_10263_ATP_Olambda__59,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_fl(Uu2),Uua) = aa_int_int(times_times_int(aa_nat_int(Uu2,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_59
tff(fact_10264_ATP_Olambda__60,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mv(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_60
tff(fact_10265_ATP_Olambda__61,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mt(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,Uua),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_61
tff(fact_10266_ATP_Olambda__62,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mu(Uu2),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_62
tff(fact_10267_ATP_Olambda__63,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_mp(Uu2),Uua) = minus_8373710615458151222nteger(aa_nat_Code_integer(Uu2,Uua),aa_nat_Code_integer(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_63
tff(fact_10268_ATP_Olambda__64,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mo(Uu2),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_64
tff(fact_10269_ATP_Olambda__65,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ms(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_65
tff(fact_10270_ATP_Olambda__66,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mq(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,Uua),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_66
tff(fact_10271_ATP_Olambda__67,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mr(Uu2),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_67
tff(fact_10272_ATP_Olambda__68,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ob(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_68
tff(fact_10273_ATP_Olambda__69,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nz(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,Uua),aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_69
tff(fact_10274_ATP_Olambda__70,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_oa(Uu2),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_70
tff(fact_10275_ATP_Olambda__71,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_nr(Uu2),Uua) = aa_real_complex(complex2(aa_complex_real(Uu2,Uua)),zero_zero_real) ).

% ATP.lambda_71
tff(fact_10276_ATP_Olambda__72,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_nq(Uu2),Uua) = aa_real_complex(complex2(aa_nat_real(Uu2,Uua)),zero_zero_real) ).

% ATP.lambda_72
tff(fact_10277_ATP_Olambda__73,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kx(Uu2),Uua))
    <=> ( aa_com4217951221296830281nteger(Uu2,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_73
tff(fact_10278_ATP_Olambda__74,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kq(Uu2),Uua))
    <=> ( aa_complex_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_74
tff(fact_10279_ATP_Olambda__75,axiom,
    ! [Uu2: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ks(Uu2),Uua))
    <=> ( aa_complex_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_75
tff(fact_10280_ATP_Olambda__76,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ku(Uu2),Uua))
    <=> ( aa_complex_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_76
tff(fact_10281_ATP_Olambda__77,axiom,
    ! [Uu2: fun_complex_int,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_kv(Uu2),Uua))
    <=> ( aa_complex_int(Uu2,Uua) = zero_zero_int ) ) ).

% ATP.lambda_77
tff(fact_10282_ATP_Olambda__78,axiom,
    ! [Uu2: fun_int_Code_integer,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kw(Uu2),Uua))
    <=> ( aa_int_Code_integer(Uu2,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_78
tff(fact_10283_ATP_Olambda__79,axiom,
    ! [Uu2: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ko(Uu2),Uua))
    <=> ( aa_int_complex(Uu2,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_79
tff(fact_10284_ATP_Olambda__80,axiom,
    ! [Uu2: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kp(Uu2),Uua))
    <=> ( aa_int_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_80
tff(fact_10285_ATP_Olambda__81,axiom,
    ! [Uu2: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kr(Uu2),Uua))
    <=> ( aa_int_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_81
tff(fact_10286_ATP_Olambda__82,axiom,
    ! [Uu2: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_kt(Uu2),Uua))
    <=> ( aa_int_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_82
tff(fact_10287_ATP_Olambda__83,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_su(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_sd(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_83
tff(fact_10288_ATP_Olambda__84,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_st(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_sd(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua))) ).

% ATP.lambda_84
tff(fact_10289_ATP_Olambda__85,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_ub(Uu2),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu2)),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_num_nat(numeral_numeral_nat,Uu2))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_85
tff(fact_10290_ATP_Olambda__86,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_re,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu2)) ).

% ATP.lambda_86
tff(fact_10291_ATP_Olambda__87,axiom,
    ! [Uu2: bool,Uua: nat] : aa_nat_nat(aa_bool_fun_nat_nat(aTP_Lamm_tw,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_bool_nat(zero_n2687167440665602831ol_nat,Uu2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,bit0(one)))) ).

% ATP.lambda_87
tff(fact_10292_ATP_Olambda__88,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pv,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_88
tff(fact_10293_ATP_Olambda__89,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pn,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_89
tff(fact_10294_ATP_Olambda__90,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qm,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)) ).

% ATP.lambda_90
tff(fact_10295_ATP_Olambda__91,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sp,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_91
tff(fact_10296_ATP_Olambda__92,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_qt,Uu2),Uua) = aa_int_int(plus_plus_int(Uu2),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).

% ATP.lambda_92
tff(fact_10297_ATP_Olambda__93,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_93
tff(fact_10298_ATP_Olambda__94,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sk,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_94
tff(fact_10299_ATP_Olambda__95,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tt,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu2)) ) ).

% ATP.lambda_95
tff(fact_10300_ATP_Olambda__96,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rf,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu2) ).

% ATP.lambda_96
tff(fact_10301_ATP_Olambda__97,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aTP_Lamm_nv,Uu2),Uua))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Uua),Uu2)) ) ).

% ATP.lambda_97
tff(fact_10302_ATP_Olambda__98,axiom,
    ! [Uu2: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_nw,Uu2),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu2)) ) ).

% ATP.lambda_98
tff(fact_10303_ATP_Olambda__99,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_ns,Uu2),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu2)) ) ).

% ATP.lambda_99
tff(fact_10304_ATP_Olambda__100,axiom,
    ! [Uu2: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_nt,Uu2),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu2)) ) ).

% ATP.lambda_100
tff(fact_10305_ATP_Olambda__101,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_dw,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu2)) ) ).

% ATP.lambda_101
tff(fact_10306_ATP_Olambda__102,axiom,
    ! [Uu2: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_nu,Uu2),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu2)) ) ).

% ATP.lambda_102
tff(fact_10307_ATP_Olambda__103,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_at,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu2) ).

% ATP.lambda_103
tff(fact_10308_ATP_Olambda__104,axiom,
    ! [Uu2: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_au(Uu2),Uua) = aa_rat_rat(times_times_rat(Uua),Uu2) ).

% ATP.lambda_104
tff(fact_10309_ATP_Olambda__105,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_av,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ).

% ATP.lambda_105
tff(fact_10310_ATP_Olambda__106,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_aw(Uu2),Uua) = aa_int_int(times_times_int(Uua),Uu2) ).

% ATP.lambda_106
tff(fact_10311_ATP_Olambda__107,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu2) ).

% ATP.lambda_107
tff(fact_10312_ATP_Olambda__108,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rk,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ).

% ATP.lambda_108
tff(fact_10313_ATP_Olambda__109,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_rh(Uu2),Uua) = plus_p5714425477246183910nteger(Uua,Uu2) ).

% ATP.lambda_109
tff(fact_10314_ATP_Olambda__110,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ty,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ).

% ATP.lambda_110
tff(fact_10315_ATP_Olambda__111,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_rg(Uu2),Uua) = aa_int_int(plus_plus_int(Uua),Uu2) ).

% ATP.lambda_111
tff(fact_10316_ATP_Olambda__112,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu2) ).

% ATP.lambda_112
tff(fact_10317_ATP_Olambda__113,axiom,
    ! [Uu2: complex,Uua: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_af,Uu2),Uua))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,Uua),Uu2)) ) ).

% ATP.lambda_113
tff(fact_10318_ATP_Olambda__114,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ah,Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu2)) ) ).

% ATP.lambda_114
tff(fact_10319_ATP_Olambda__115,axiom,
    ! [Uu2: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ag,Uu2),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu2)) ) ).

% ATP.lambda_115
tff(fact_10320_ATP_Olambda__116,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qc,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu2) ).

% ATP.lambda_116
tff(fact_10321_ATP_Olambda__117,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sj,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ).

% ATP.lambda_117
tff(fact_10322_ATP_Olambda__118,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aTP_Lamm_aa(Uu2),Uua))
    <=> member_VEBT_VEBT(Uua,Uu2) ) ).

% ATP.lambda_118
tff(fact_10323_ATP_Olambda__119,axiom,
    ! [Uu2: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ae(Uu2),Uua))
    <=> member5262025264175285858nt_int(Uua,Uu2) ) ).

% ATP.lambda_119
tff(fact_10324_ATP_Olambda__120,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_ad(Uu2),Uua))
    <=> member_complex(Uua,Uu2) ) ).

% ATP.lambda_120
tff(fact_10325_ATP_Olambda__121,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_a(Uu2),Uua))
    <=> pp(aa_set_real_bool(member_real(Uua),Uu2)) ) ).

% ATP.lambda_121
tff(fact_10326_ATP_Olambda__122,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ab(Uu2),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu2)) ) ).

% ATP.lambda_122
tff(fact_10327_ATP_Olambda__123,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ac(Uu2),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu2)) ) ).

% ATP.lambda_123
tff(fact_10328_ATP_Olambda__124,axiom,
    ! [Uu2: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_de(Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Pro6801748904974264397nt_nat(Uu2,Uua))) ) ).

% ATP.lambda_124
tff(fact_10329_ATP_Olambda__125,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_dk(Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu2,Uua))) ) ).

% ATP.lambda_125
tff(fact_10330_ATP_Olambda__126,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_dg(Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu2,Uua))) ) ).

% ATP.lambda_126
tff(fact_10331_ATP_Olambda__127,axiom,
    ! [Uu2: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_di(Uu2),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(Uu2,Uua))) ) ).

% ATP.lambda_127
tff(fact_10332_ATP_Olambda__128,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pk(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_128
tff(fact_10333_ATP_Olambda__129,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pj(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)) ).

% ATP.lambda_129
tff(fact_10334_ATP_Olambda__130,axiom,
    ! [Uu2: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_th(Uu2),Uua))
    <=> pp(aa_real_bool(Uu2,aa_real_real(inverse_inverse_real,Uua))) ) ).

% ATP.lambda_130
tff(fact_10335_ATP_Olambda__131,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fg(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_131
tff(fact_10336_ATP_Olambda__132,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_td(Uu2),Uua))
    <=> pp(aa_nat_bool(Uu2,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_132
tff(fact_10337_ATP_Olambda__133,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lr(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_133
tff(fact_10338_ATP_Olambda__134,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ky(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_134
tff(fact_10339_ATP_Olambda__135,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ls(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_135
tff(fact_10340_ATP_Olambda__136,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_uh(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_136
tff(fact_10341_ATP_Olambda__137,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ud(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_137
tff(fact_10342_ATP_Olambda__138,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_rz(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ry(Uu2),Uua)) ).

% ATP.lambda_138
tff(fact_10343_ATP_Olambda__139,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_qn(Uu2),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).

% ATP.lambda_139
tff(fact_10344_ATP_Olambda__140,axiom,
    ! [Uu2: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ta(Uu2),Uua))
    <=> ( aa_real_real(Uu2,Uua) != zero_zero_real ) ) ).

% ATP.lambda_140
tff(fact_10345_ATP_Olambda__141,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ax,Uu2),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)))) ) ).

% ATP.lambda_141
tff(fact_10346_ATP_Olambda__142,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_142
tff(fact_10347_ATP_Olambda__143,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sg(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_143
tff(fact_10348_ATP_Olambda__144,axiom,
    ! [Uu2: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_bz(Uu2),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu2,Uua)) ).

% ATP.lambda_144
tff(fact_10349_ATP_Olambda__145,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_hw(Uu2),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu2,Uua)) ).

% ATP.lambda_145
tff(fact_10350_ATP_Olambda__146,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_complex(aTP_Lamm_hz(Uu2),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_146
tff(fact_10351_ATP_Olambda__147,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ca(Uu2),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_147
tff(fact_10352_ATP_Olambda__148,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ia(Uu2),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_148
tff(fact_10353_ATP_Olambda__149,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_hy(Uu2),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_149
tff(fact_10354_ATP_Olambda__150,axiom,
    ! [Uu2: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_hx(Uu2),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu2,Uua)) ).

% ATP.lambda_150
tff(fact_10355_ATP_Olambda__151,axiom,
    ! [Uu2: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_ib(Uu2),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu2,Uua)) ).

% ATP.lambda_151
tff(fact_10356_ATP_Olambda__152,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_ic(Uu2),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_152
tff(fact_10357_ATP_Olambda__153,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_complex(aTP_Lamm_if(Uu2),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_153
tff(fact_10358_ATP_Olambda__154,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_id(Uu2),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_154
tff(fact_10359_ATP_Olambda__155,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_ie(Uu2),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_155
tff(fact_10360_ATP_Olambda__156,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ig(Uu2),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_156
tff(fact_10361_ATP_Olambda__157,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ja(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_157
tff(fact_10362_ATP_Olambda__158,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_jb(Uu2),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_158
tff(fact_10363_ATP_Olambda__159,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ss(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_159
tff(fact_10364_ATP_Olambda__160,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_sb(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ).

% ATP.lambda_160
tff(fact_10365_ATP_Olambda__161,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qa(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one))))),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,bit0(one))))) ).

% ATP.lambda_161
tff(fact_10366_ATP_Olambda__162,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pi(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uub)),aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_162
tff(fact_10367_ATP_Olambda__163,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_qr(Uu2),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu2)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Uua),Uub)) ).

% ATP.lambda_163
tff(fact_10368_ATP_Olambda__164,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_qp(Uu2),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu2)),Uub)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu2)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),Uua)),Uub)) ).

% ATP.lambda_164
tff(fact_10369_ATP_Olambda__165,axiom,
    ! [Uu2: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qq(Uu2),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),Uub),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Uua)),one_one_int),aa_int_int(minus_minus_int(Uub),aa_num_int(numeral_numeral_int,Uu2))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Uua),Uub)) ).

% ATP.lambda_165
tff(fact_10370_ATP_Olambda__166,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ev(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_set_nat_bool(member_nat(Uub),Uu2),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_166
tff(fact_10371_ATP_Olambda__167,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ew(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu2)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_167
tff(fact_10372_ATP_Olambda__168,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ex(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(member_nat(Uub),Uu2)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_168
tff(fact_10373_ATP_Olambda__169,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ey(Uu2),Uua),Uub) = aa_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu2),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_169
tff(fact_10374_ATP_Olambda__170,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_ir(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu2),Uub),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_170
tff(fact_10375_ATP_Olambda__171,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_iv(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu2),Uub)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_171
tff(fact_10376_ATP_Olambda__172,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_iz(Uu2),Uua),Uub) = if_rat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uu2),Uub),aa_VEBT_VEBT_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_172
tff(fact_10377_ATP_Olambda__173,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ix(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu2),Uub)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_173
tff(fact_10378_ATP_Olambda__174,axiom,
    ! [Uu2: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_iq(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu2),Uub),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_174
tff(fact_10379_ATP_Olambda__175,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_iu(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu2),Uub)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_175
tff(fact_10380_ATP_Olambda__176,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iy(Uu2),Uua),Uub) = if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu2),Uub),aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_176
tff(fact_10381_ATP_Olambda__177,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_is(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu2),Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_177
tff(fact_10382_ATP_Olambda__178,axiom,
    ! [Uu2: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_it(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu2),Uub),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_178
tff(fact_10383_ATP_Olambda__179,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_iw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu2),Uub)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_179
tff(fact_10384_ATP_Olambda__180,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun8982142231764154876omplex(aTP_Lamm_ii(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu2),aa_VEBT_VEBT_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_180
tff(fact_10385_ATP_Olambda__181,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_il(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu2)),aa_VEBT_VEBT_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_181
tff(fact_10386_ATP_Olambda__182,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ip(Uu2),Uua),Uub) = if_rat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uub),Uu2),aa_VEBT_VEBT_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_182
tff(fact_10387_ATP_Olambda__183,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_in(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu2)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_183
tff(fact_10388_ATP_Olambda__184,axiom,
    ! [Uu2: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ih(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu2),aa_real_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_184
tff(fact_10389_ATP_Olambda__185,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ik(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu2)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_185
tff(fact_10390_ATP_Olambda__186,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_io(Uu2),Uua),Uub) = if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu2),aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_186
tff(fact_10391_ATP_Olambda__187,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ei(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu2),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_187
tff(fact_10392_ATP_Olambda__188,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ej(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu2)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_188
tff(fact_10393_ATP_Olambda__189,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ek(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu2)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_189
tff(fact_10394_ATP_Olambda__190,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_el(Uu2),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu2),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_190
tff(fact_10395_ATP_Olambda__191,axiom,
    ! [Uu2: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ij(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu2),aa_int_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_191
tff(fact_10396_ATP_Olambda__192,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_im(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu2)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_192
tff(fact_10397_ATP_Olambda__193,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_er(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uu2,Uub),aa_nat_complex(Uua,Uub)),zero_zero_complex) ).

% ATP.lambda_193
tff(fact_10398_ATP_Olambda__194,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_es(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uu2,Uub)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_194
tff(fact_10399_ATP_Olambda__195,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_et(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uu2,Uub)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_195
tff(fact_10400_ATP_Olambda__196,axiom,
    ! [Uu2: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_eu(Uu2),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu2,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_196
tff(fact_10401_ATP_Olambda__197,axiom,
    ! [Uu2: fun_VE5569070535936029429omplex,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun6792494735270171281omplex(aTP_Lamm_kc(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_complex(Uu2,Uub)),zero_zero_complex) ).

% ATP.lambda_197
tff(fact_10402_ATP_Olambda__198,axiom,
    ! [Uu2: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun2072160908118572431T_real(aTP_Lamm_ki(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(Uua,Uub)),aa_VEBT_VEBT_real(Uu2,Uub)),zero_zero_real) ).

% ATP.lambda_198
tff(fact_10403_ATP_Olambda__199,axiom,
    ! [Uu2: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun3193099451372820155BT_rat(aTP_Lamm_kn(Uu2),Uua),Uub) = if_rat(aa_VEBT_VEBT_bool(Uua,Uub),aa_VEBT_VEBT_rat(Uu2,Uub),zero_zero_rat) ).

% ATP.lambda_199
tff(fact_10404_ATP_Olambda__200,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_kl(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu2,Uub)),zero_zero_real) ).

% ATP.lambda_200
tff(fact_10405_ATP_Olambda__201,axiom,
    ! [Uu2: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_ka(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu2,Uub)),zero_zero_complex) ).

% ATP.lambda_201
tff(fact_10406_ATP_Olambda__202,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_kh(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu2,Uub)),zero_zero_real) ).

% ATP.lambda_202
tff(fact_10407_ATP_Olambda__203,axiom,
    ! [Uu2: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_km(Uu2),Uua),Uub) = if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu2,Uub),zero_zero_rat) ).

% ATP.lambda_203
tff(fact_10408_ATP_Olambda__204,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_ke(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu2,Uub)),zero_zero_complex) ).

% ATP.lambda_204
tff(fact_10409_ATP_Olambda__205,axiom,
    ! [Uu2: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_kg(Uu2),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu2,Uub)),zero_zero_complex) ).

% ATP.lambda_205
tff(fact_10410_ATP_Olambda__206,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_kj(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu2,Uub)),zero_zero_real) ).

% ATP.lambda_206
tff(fact_10411_ATP_Olambda__207,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_tb(Uu2),Uua),Uub))
    <=> has_fi5821293074295781190e_real(Uu2,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_207
tff(fact_10412_ATP_Olambda__208,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_sa(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_208
tff(fact_10413_ATP_Olambda__209,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mm(Uu2),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_209
tff(fact_10414_ATP_Olambda__210,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mj(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu2,Uub),Uua) ).

% ATP.lambda_210
tff(fact_10415_ATP_Olambda__211,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mn(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mm(Uu2),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_211
tff(fact_10416_ATP_Olambda__212,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mk(Uu2),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mj(Uu2),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_212
tff(fact_10417_ATP_Olambda__213,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_tp(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_to(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_213
tff(fact_10418_ATP_Olambda__214,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_tn(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_tm(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_214
tff(fact_10419_ATP_Olambda__215,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_tl(Uu2),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_tk(Uua),Uub)),quotient_of(Uu2))) ) ).

% ATP.lambda_215
tff(fact_10420_ATP_Olambda__216,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rp(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_216
tff(fact_10421_ATP_Olambda__217,axiom,
    ! [Uu2: real,Uua: fun_na5509613867492920147r_real,Uub: nat] : aa_nat_real(aa_fun5144707104869965066t_real(aTP_Lamm_pt(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_Code_integer_real(aa_nat8538992517141809556r_real(Uua,Uub),zero_z3403309356797280102nteger)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_217
tff(fact_10422_ATP_Olambda__218,axiom,
    ! [Uu2: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_po(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_218
tff(fact_10423_ATP_Olambda__219,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_pp(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_219
tff(fact_10424_ATP_Olambda__220,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_pq(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_220
tff(fact_10425_ATP_Olambda__221,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_pr(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_221
tff(fact_10426_ATP_Olambda__222,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_ps(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_222
tff(fact_10427_ATP_Olambda__223,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jd(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_223
tff(fact_10428_ATP_Olambda__224,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jf(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_224
tff(fact_10429_ATP_Olambda__225,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_je(Uu2),Uua),Uub) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_225
tff(fact_10430_ATP_Olambda__226,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rx(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_226
tff(fact_10431_ATP_Olambda__227,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pu(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_227
tff(fact_10432_ATP_Olambda__228,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cy(Uu2),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).

% ATP.lambda_228
tff(fact_10433_ATP_Olambda__229,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_cz(Uu2),Uua),Uub))
    <=> ( ord_le211207098394363844omplex(set_complex2(Uub),Uu2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_229
tff(fact_10434_ATP_Olambda__230,axiom,
    ! [Uu2: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_da(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Uub)),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_real(Uub)),Uua)) ) ) ).

% ATP.lambda_230
tff(fact_10435_ATP_Olambda__231,axiom,
    ! [Uu2: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_db(Uu2),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu2)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).

% ATP.lambda_231
tff(fact_10436_ATP_Olambda__232,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_dd(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_232
tff(fact_10437_ATP_Olambda__233,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_dc(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).

% ATP.lambda_233
tff(fact_10438_ATP_Olambda__234,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_cs(Uu2),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu2)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_234
tff(fact_10439_ATP_Olambda__235,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_ct(Uu2),Uua),Uub))
    <=> ( ord_le211207098394363844omplex(set_complex2(Uub),Uu2)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_235
tff(fact_10440_ATP_Olambda__236,axiom,
    ! [Uu2: set_real,Uua: nat,Uub: list_real] :
      ( pp(aa_list_real_bool(aa_nat4570152432052498854l_bool(aTP_Lamm_cu(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Uub)),Uu2))
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_236
tff(fact_10441_ATP_Olambda__237,axiom,
    ! [Uu2: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_cv(Uu2),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu2)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_237
tff(fact_10442_ATP_Olambda__238,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_cx(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu2))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_238
tff(fact_10443_ATP_Olambda__239,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_cw(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu2))
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_239
tff(fact_10444_ATP_Olambda__240,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tu(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_240
tff(fact_10445_ATP_Olambda__241,axiom,
    ! [Uu2: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_rd(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Uua,Uub)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu2),Uub)) ).

% ATP.lambda_241
tff(fact_10446_ATP_Olambda__242,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tv(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_242
tff(fact_10447_ATP_Olambda__243,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qf(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uub)) ).

% ATP.lambda_243
tff(fact_10448_ATP_Olambda__244,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_qu(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_244
tff(fact_10449_ATP_Olambda__245,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ea(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_245
tff(fact_10450_ATP_Olambda__246,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_qv(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_246
tff(fact_10451_ATP_Olambda__247,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ed(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_247
tff(fact_10452_ATP_Olambda__248,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ee(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_248
tff(fact_10453_ATP_Olambda__249,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_eb(Uu2),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu2),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_249
tff(fact_10454_ATP_Olambda__250,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_az(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu2),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_250
tff(fact_10455_ATP_Olambda__251,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ay(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,Uu2),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_251
tff(fact_10456_ATP_Olambda__252,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_bool,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun4462450708475455754T_bool(aTP_Lamm_kb(Uu2),Uua),Uub))
    <=> ( member_VEBT_VEBT(Uub,Uu2)
        & pp(aa_VEBT_VEBT_bool(Uua,Uub)) ) ) ).

% ATP.lambda_252
tff(fact_10457_ATP_Olambda__253,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_kk(Uu2),Uua),Uub))
    <=> ( member_complex(Uub,Uu2)
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_253
tff(fact_10458_ATP_Olambda__254,axiom,
    ! [Uu2: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_jz(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_254
tff(fact_10459_ATP_Olambda__255,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_kd(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_255
tff(fact_10460_ATP_Olambda__256,axiom,
    ! [Uu2: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_kf(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_256
tff(fact_10461_ATP_Olambda__257,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aTP_Lamm_he(Uu2),Uua),Uub))
    <=> ( member_VEBT_VEBT(Uub,Uu2)
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_257
tff(fact_10462_ATP_Olambda__258,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_ho(Uu2),Uua),Uub))
    <=> ( member_VEBT_VEBT(Uub,Uu2)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_258
tff(fact_10463_ATP_Olambda__259,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_hk(Uu2),Uua),Uub))
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_259
tff(fact_10464_ATP_Olambda__260,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_hu(Uu2),Uua),Uub))
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_260
tff(fact_10465_ATP_Olambda__261,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_hc(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_261
tff(fact_10466_ATP_Olambda__262,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_hm(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_262
tff(fact_10467_ATP_Olambda__263,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_hg(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_263
tff(fact_10468_ATP_Olambda__264,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_hq(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_264
tff(fact_10469_ATP_Olambda__265,axiom,
    ! [Uu2: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_hi(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_265
tff(fact_10470_ATP_Olambda__266,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_hs(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_266
tff(fact_10471_ATP_Olambda__267,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun437439898845168857T_bool(aTP_Lamm_gk(Uu2),Uua),Uub))
    <=> ( member_VEBT_VEBT(Uub,Uu2)
        & ( aa_VEBT_VEBT_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_267
tff(fact_10472_ATP_Olambda__268,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aTP_Lamm_gu(Uu2),Uua),Uub))
    <=> ( member_VEBT_VEBT(Uub,Uu2)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_268
tff(fact_10473_ATP_Olambda__269,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_assn,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1925668587588942193x_bool(aTP_Lamm_gq(Uu2),Uua),Uub))
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_269
tff(fact_10474_ATP_Olambda__270,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_ha(Uu2),Uua),Uub))
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_270
tff(fact_10475_ATP_Olambda__271,axiom,
    ! [Uu2: set_real,Uua: fun_real_assn,Uub: real] :
      ( pp(aa_real_bool(aa_fun159070158279343989l_bool(aTP_Lamm_gi(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & ( aa_real_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_271
tff(fact_10476_ATP_Olambda__272,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_gs(Uu2),Uua),Uub))
    <=> ( pp(aa_set_real_bool(member_real(Uub),Uu2))
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_272
tff(fact_10477_ATP_Olambda__273,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1272683333798124589t_bool(aTP_Lamm_gm(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_273
tff(fact_10478_ATP_Olambda__274,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_gw(Uu2),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu2))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_274
tff(fact_10479_ATP_Olambda__275,axiom,
    ! [Uu2: set_int,Uua: fun_int_assn,Uub: int] :
      ( pp(aa_int_bool(aa_fun1414614411059211637t_bool(aTP_Lamm_go(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_275
tff(fact_10480_ATP_Olambda__276,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_gy(Uu2),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu2))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_276
tff(fact_10481_ATP_Olambda__277,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_bp(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),vEBT_vebt_mint(Uu2)))) ).

% ATP.lambda_277
tff(fact_10482_ATP_Olambda__278,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_bq(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),vEBT_vebt_maxt(Uu2)))) ).

% ATP.lambda_278
tff(fact_10483_ATP_Olambda__279,axiom,
    ! [Uu2: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_lis3169797078229555565i_assn(aTP_Lamm_pm(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Uu2,Uua)),aa_bool_assn(pure_assn,fequal1389750275112723248_VEBTi(Uub,Uua))) ).

% ATP.lambda_279
tff(fact_10484_ATP_Olambda__280,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_rj(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu2)),Uua)) ) ).

% ATP.lambda_280
tff(fact_10485_ATP_Olambda__281,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_ec(Uu2),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).

% ATP.lambda_281
tff(fact_10486_ATP_Olambda__282,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_qo(Uu2),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_282
tff(fact_10487_ATP_Olambda__283,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fm(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_283
tff(fact_10488_ATP_Olambda__284,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_284
tff(fact_10489_ATP_Olambda__285,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_sr(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) ).

% ATP.lambda_285
tff(fact_10490_ATP_Olambda__286,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ry(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_286
tff(fact_10491_ATP_Olambda__287,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fj(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_287
tff(fact_10492_ATP_Olambda__288,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fh(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_288
tff(fact_10493_ATP_Olambda__289,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_289
tff(fact_10494_ATP_Olambda__290,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cr(Uu2),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu2,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_290
tff(fact_10495_ATP_Olambda__291,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_291
tff(fact_10496_ATP_Olambda__292,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu2,Uub)) ).

% ATP.lambda_292
tff(fact_10497_ATP_Olambda__293,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ni(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_293
tff(fact_10498_ATP_Olambda__294,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ng(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_294
tff(fact_10499_ATP_Olambda__295,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nj(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_295
tff(fact_10500_ATP_Olambda__296,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_nh(Uu2),Uua),Uub) = aa_int_int(times_times_int(aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_296
tff(fact_10501_ATP_Olambda__297,axiom,
    ! [Uu2: fun_VE9219535225822561060nteger,Uua: fun_VE9219535225822561060nteger,Uub: vEBT_VEBT] : aa_VEB5935388676523804373nteger(aa_fun2400605388621188028nteger(aTP_Lamm_ma(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_VEB5935388676523804373nteger(Uua,Uub)),aa_VEB5935388676523804373nteger(Uu2,Uub)) ).

% ATP.lambda_297
tff(fact_10502_ATP_Olambda__298,axiom,
    ! [Uu2: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_mf(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(Uua,Uub)),aa_VEBT_VEBT_rat(Uu2,Uub)) ).

% ATP.lambda_298
tff(fact_10503_ATP_Olambda__299,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_mc(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu2,Uub)) ).

% ATP.lambda_299
tff(fact_10504_ATP_Olambda__300,axiom,
    ! [Uu2: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_mh(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu2,Uub)) ).

% ATP.lambda_300
tff(fact_10505_ATP_Olambda__301,axiom,
    ! [Uu2: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_lz(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu2,Uub)) ).

% ATP.lambda_301
tff(fact_10506_ATP_Olambda__302,axiom,
    ! [Uu2: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_me(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu2,Uub)) ).

% ATP.lambda_302
tff(fact_10507_ATP_Olambda__303,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ly(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu2,Uub)) ).

% ATP.lambda_303
tff(fact_10508_ATP_Olambda__304,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_md(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu2,Uub)) ).

% ATP.lambda_304
tff(fact_10509_ATP_Olambda__305,axiom,
    ! [Uu2: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_mb(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu2,Uub)) ).

% ATP.lambda_305
tff(fact_10510_ATP_Olambda__306,axiom,
    ! [Uu2: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_mg(Uu2),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu2,Uub)) ).

% ATP.lambda_306
tff(fact_10511_ATP_Olambda__307,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sf(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_307
tff(fact_10512_ATP_Olambda__308,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_jg(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_308
tff(fact_10513_ATP_Olambda__309,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fb(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_309
tff(fact_10514_ATP_Olambda__310,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fc(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_310
tff(fact_10515_ATP_Olambda__311,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_fd(Uu2),Uua),Uub) = aa_int_int(plus_plus_int(aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_311
tff(fact_10516_ATP_Olambda__312,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_jh(Uu2),Uua),Uub) = aa_int_int(plus_plus_int(aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_312
tff(fact_10517_ATP_Olambda__313,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ro(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_313
tff(fact_10518_ATP_Olambda__314,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jx(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_314
tff(fact_10519_ATP_Olambda__315,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jy(Uu2),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_315
tff(fact_10520_ATP_Olambda__316,axiom,
    ! [Uu2: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ji(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_316
tff(fact_10521_ATP_Olambda__317,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ep(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_317
tff(fact_10522_ATP_Olambda__318,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eq(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_318
tff(fact_10523_ATP_Olambda__319,axiom,
    ! [Uu2: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_bw(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(Uu2,Uub)),Uua) ).

% ATP.lambda_319
tff(fact_10524_ATP_Olambda__320,axiom,
    ! [Uu2: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_js(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_320
tff(fact_10525_ATP_Olambda__321,axiom,
    ! [Uu2: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_bu(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(aa_option_nat_assn(Uu2,Uub)),Uua) ).

% ATP.lambda_321
tff(fact_10526_ATP_Olambda__322,axiom,
    ! [Uu2: fun_bool_assn,Uua: assn,Uub: bool] : aa_bool_assn(aa_ass2376637134808068204l_assn(aTP_Lamm_bt(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(aa_bool_assn(Uu2,Uub)),Uua) ).

% ATP.lambda_322
tff(fact_10527_ATP_Olambda__323,axiom,
    ! [Uu2: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_bv(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(aa_nat_assn(Uu2,Uub)),Uua) ).

% ATP.lambda_323
tff(fact_10528_ATP_Olambda__324,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fe(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_324
tff(fact_10529_ATP_Olambda__325,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jr(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_325
tff(fact_10530_ATP_Olambda__326,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jt(Uu2),Uua),Uub) = aa_int_int(times_times_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_326
tff(fact_10531_ATP_Olambda__327,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_oe(Uu2),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_327
tff(fact_10532_ATP_Olambda__328,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_of(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_328
tff(fact_10533_ATP_Olambda__329,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_oc(Uu2),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu2,Uub),Uua) ).

% ATP.lambda_329
tff(fact_10534_ATP_Olambda__330,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_od(Uu2),Uua),Uub) = aa_int_int(minus_minus_int(aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_330
tff(fact_10535_ATP_Olambda__331,axiom,
    ! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rl(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_331
tff(fact_10536_ATP_Olambda__332,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sw(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ).

% ATP.lambda_332
tff(fact_10537_ATP_Olambda__333,axiom,
    ! [Uu2: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_333
tff(fact_10538_ATP_Olambda__334,axiom,
    ! [Uu2: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_fr(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_334
tff(fact_10539_ATP_Olambda__335,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_oh(Uu2),Uua),Uub) = groups6591440286371151544t_real(Uu2,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_335
tff(fact_10540_ATP_Olambda__336,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_og(Uu2),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu2),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uua))) ).

% ATP.lambda_336
tff(fact_10541_ATP_Olambda__337,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_nc(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_337
tff(fact_10542_ATP_Olambda__338,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ne(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_338
tff(fact_10543_ATP_Olambda__339,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_na(Uu2),Uua),Uub) = plus_plus_rat(Uu2,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_339
tff(fact_10544_ATP_Olambda__340,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nd(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_340
tff(fact_10545_ATP_Olambda__341,axiom,
    ! [Uu2: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_nb(Uu2),Uua),Uub) = aa_int_int(plus_plus_int(Uu2),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_341
tff(fact_10546_ATP_Olambda__342,axiom,
    ! [Uu2: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_pe(Uu2),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_342
tff(fact_10547_ATP_Olambda__343,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pd(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_343
tff(fact_10548_ATP_Olambda__344,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ph(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_344
tff(fact_10549_ATP_Olambda__345,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_pf(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_345
tff(fact_10550_ATP_Olambda__346,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pg(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_346
tff(fact_10551_ATP_Olambda__347,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sl(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_347
tff(fact_10552_ATP_Olambda__348,axiom,
    ! [Uu2: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_lb(Uu2),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_348
tff(fact_10553_ATP_Olambda__349,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_la(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_349
tff(fact_10554_ATP_Olambda__350,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_le(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_350
tff(fact_10555_ATP_Olambda__351,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_lc(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_351
tff(fact_10556_ATP_Olambda__352,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ld(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_352
tff(fact_10557_ATP_Olambda__353,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nf(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_353
tff(fact_10558_ATP_Olambda__354,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tg(Uu2),Uua),Uub))
    <=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu2,Uua))) ) ).

% ATP.lambda_354
tff(fact_10559_ATP_Olambda__355,axiom,
    ! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_jv(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(Uu2),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_355
tff(fact_10560_ATP_Olambda__356,axiom,
    ! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_en(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(Uu2),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_356
tff(fact_10561_ATP_Olambda__357,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_eo(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_357
tff(fact_10562_ATP_Olambda__358,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ju(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_358
tff(fact_10563_ATP_Olambda__359,axiom,
    ! [Uu2: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_jw(Uu2),Uua),Uub) = aa_int_int(times_times_int(Uu2),aa_int_int(Uua,Uub)) ).

% ATP.lambda_359
tff(fact_10564_ATP_Olambda__360,axiom,
    ! [Uu2: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_cn(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_VEBT_VEBTi_assn(Uu2,Uub)) ).

% ATP.lambda_360
tff(fact_10565_ATP_Olambda__361,axiom,
    ! [Uu2: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_cl(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_option_nat_assn(Uu2,Uub)) ).

% ATP.lambda_361
tff(fact_10566_ATP_Olambda__362,axiom,
    ! [Uu2: fun_bool_assn,Uua: assn,Uub: bool] : aa_bool_assn(aa_ass2376637134808068204l_assn(aTP_Lamm_ck(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_bool_assn(Uu2,Uub)) ).

% ATP.lambda_362
tff(fact_10567_ATP_Olambda__363,axiom,
    ! [Uu2: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_cm(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_nat_assn(Uu2,Uub)) ).

% ATP.lambda_363
tff(fact_10568_ATP_Olambda__364,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ff(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu2,Uub)) ).

% ATP.lambda_364
tff(fact_10569_ATP_Olambda__365,axiom,
    ! [Uu2: assn,Uua: vEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(aTP_Lamm_ce(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uu2),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Uub),Uua))) ).

% ATP.lambda_365
tff(fact_10570_ATP_Olambda__366,axiom,
    ! [Uu2: assn,Uua: option_nat,Uub: option_nat] : aa_option_nat_assn(aa_opt3861602675778688675t_assn(aTP_Lamm_cc(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uu2),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uub),Uua))) ).

% ATP.lambda_366
tff(fact_10571_ATP_Olambda__367,axiom,
    ! [Uu2: assn,Uua: bool,Uub: bool] : aa_bool_assn(aa_boo2938360491098410167l_assn(aTP_Lamm_cb(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uu2),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uub),Uua))) ).

% ATP.lambda_367
tff(fact_10572_ATP_Olambda__368,axiom,
    ! [Uu2: assn,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_cd(Uu2),Uua),Uub) = aa_assn_assn(times_times_assn(Uu2),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uua))) ).

% ATP.lambda_368
tff(fact_10573_ATP_Olambda__369,axiom,
    ! [Uu2: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(aTP_Lamm_dm(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Pro4501881432599481050t_bool(fequal6010348034681699726nt_int(Uub),Uua)),aa_nat_nat(suc,aa_Pro6801748904974264397nt_nat(Uu2,Uub))),aa_Pro6801748904974264397nt_nat(Uu2,Uub)))) ) ).

% ATP.lambda_369
tff(fact_10574_ATP_Olambda__370,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_dp(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uua)),aa_nat_nat(suc,aa_complex_nat(Uu2,Uub))),aa_complex_nat(Uu2,Uub)))) ) ).

% ATP.lambda_370
tff(fact_10575_ATP_Olambda__371,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_dn(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uua)),aa_nat_nat(suc,aa_nat_nat(Uu2,Uub))),aa_nat_nat(Uu2,Uub)))) ) ).

% ATP.lambda_371
tff(fact_10576_ATP_Olambda__372,axiom,
    ! [Uu2: fun_int_nat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_do(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uua)),aa_nat_nat(suc,aa_int_nat(Uu2,Uub))),aa_int_nat(Uu2,Uub)))) ) ).

% ATP.lambda_372
tff(fact_10577_ATP_Olambda__373,axiom,
    ! [Uu2: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr803113274115125473t_bool,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun8917503761599146496t_bool(aTP_Lamm_df(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Pro4501881432599481050t_bool(Uua,Uub)),aa_Pro6801748904974264397nt_nat(Uu2,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_373
tff(fact_10578_ATP_Olambda__374,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_dl(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(Uua,Uub)),aa_complex_nat(Uu2,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_374
tff(fact_10579_ATP_Olambda__375,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_dh(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uua,Uub)),aa_nat_nat(Uu2,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_375
tff(fact_10580_ATP_Olambda__376,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_dj(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(Uua,Uub)),aa_int_nat(Uu2,Uub)),zero_zero_nat))) ) ).

% ATP.lambda_376
tff(fact_10581_ATP_Olambda__377,axiom,
    ! [Uu2: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5604724294258851521t_bool(aTP_Lamm_dq(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro6801748904974264397nt_nat(Uu2,Uub)),aa_Pro6801748904974264397nt_nat(Uua,Uub)))) ) ).

% ATP.lambda_377
tff(fact_10582_ATP_Olambda__378,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun4400319914713427657x_bool(aTP_Lamm_dt(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uu2,Uub)),aa_complex_nat(Uua,Uub)))) ) ).

% ATP.lambda_378
tff(fact_10583_ATP_Olambda__379,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1783405840085286277t_bool(aTP_Lamm_dr(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)))) ) ).

% ATP.lambda_379
tff(fact_10584_ATP_Olambda__380,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( pp(aa_int_bool(aa_fun5627573752281863885t_bool(aTP_Lamm_ds(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu2,Uub)),aa_int_nat(Uua,Uub)))) ) ).

% ATP.lambda_380
tff(fact_10585_ATP_Olambda__381,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ny(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_381
tff(fact_10586_ATP_Olambda__382,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nx(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_382
tff(fact_10587_ATP_Olambda__383,axiom,
    ! [Uu2: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tf(Uu2),Uua),Uub))
    <=> pp(aa_real_bool(Uu2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua))) ) ).

% ATP.lambda_383
tff(fact_10588_ATP_Olambda__384,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_fq(Uu2),Uua),Uub) = aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_384
tff(fact_10589_ATP_Olambda__385,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_em(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_385
tff(fact_10590_ATP_Olambda__386,axiom,
    ! [Uu2: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_te(Uu2),Uua),Uub))
    <=> pp(aa_nat_bool(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_386
tff(fact_10591_ATP_Olambda__387,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kz(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_387
tff(fact_10592_ATP_Olambda__388,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_fk(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_388
tff(fact_10593_ATP_Olambda__389,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fi(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_389
tff(fact_10594_ATP_Olambda__390,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sq(Uu2),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua)) ).

% ATP.lambda_390
tff(fact_10595_ATP_Olambda__391,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_by(Uu2),Uua),Uub) = aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu2),Uua))) ).

% ATP.lambda_391
tff(fact_10596_ATP_Olambda__392,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_bx(Uu2),Uua),Uub) = aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu2),Uua))) ).

% ATP.lambda_392
tff(fact_10597_ATP_Olambda__393,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_lh(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu2)),aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_393
tff(fact_10598_ATP_Olambda__394,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_lj(Uu2),Uua),Uub),Uuc) = if_rat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu2),aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ).

% ATP.lambda_394
tff(fact_10599_ATP_Olambda__395,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_ll(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu2)),aa_VEBT_VEBT_nat(Uua,Uuc)),aa_VEBT_VEBT_nat(Uub,Uuc)) ).

% ATP.lambda_395
tff(fact_10600_ATP_Olambda__396,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: fun_VEBT_VEBT_int,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_ln(Uu2),Uua),Uub),Uuc) = aa_int_int(if_int(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu2),aa_VEBT_VEBT_int(Uua,Uuc)),aa_VEBT_VEBT_int(Uub,Uuc)) ).

% ATP.lambda_396
tff(fact_10601_ATP_Olambda__397,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_li(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu2)),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_397
tff(fact_10602_ATP_Olambda__398,axiom,
    ! [Uu2: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_lk(Uu2),Uua),Uub),Uuc) = if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu2),aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_398
tff(fact_10603_ATP_Olambda__399,axiom,
    ! [Uu2: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_lm(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu2)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_399
tff(fact_10604_ATP_Olambda__400,axiom,
    ! [Uu2: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_lo(Uu2),Uua),Uub),Uuc) = aa_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu2),aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_400
tff(fact_10605_ATP_Olambda__401,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_lp(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu2)),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_401
tff(fact_10606_ATP_Olambda__402,axiom,
    ! [Uu2: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_lq(Uu2),Uua),Uub),Uuc) = if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu2),aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_402
tff(fact_10607_ATP_Olambda__403,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_np(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_no(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_403
tff(fact_10608_ATP_Olambda__404,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(Uu2),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nm(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_404
tff(fact_10609_ATP_Olambda__405,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nl(Uu2),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nk(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_405
tff(fact_10610_ATP_Olambda__406,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_jq(Uu2),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_jp(Uu2),Uua),Uuc),Uub) ).

% ATP.lambda_406
tff(fact_10611_ATP_Olambda__407,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_jo(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_jn(Uu2),Uua),Uuc),Uub) ).

% ATP.lambda_407
tff(fact_10612_ATP_Olambda__408,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_jm(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_jl(Uu2),Uua),Uuc),Uub) ).

% ATP.lambda_408
tff(fact_10613_ATP_Olambda__409,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_jk(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jj(Uu2),Uua),Uuc)),Uub) ).

% ATP.lambda_409
tff(fact_10614_ATP_Olambda__410,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qk(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qj(Uu2),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_410
tff(fact_10615_ATP_Olambda__411,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ru(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_411
tff(fact_10616_ATP_Olambda__412,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rs(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_412
tff(fact_10617_ATP_Olambda__413,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rr(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_413
tff(fact_10618_ATP_Olambda__414,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rq(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_414
tff(fact_10619_ATP_Olambda__415,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_ol(Uu2),Uua),Uub),Uuc) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(Uu2,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),Uub)) ).

% ATP.lambda_415
tff(fact_10620_ATP_Olambda__416,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oj(Uu2),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uub)) ).

% ATP.lambda_416
tff(fact_10621_ATP_Olambda__417,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_or(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_417
tff(fact_10622_ATP_Olambda__418,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_on(Uu2),Uua),Uub),Uuc) = minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uu2,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_nat_rat(power_power_rat(Uu2),Uub)) ).

% ATP.lambda_418
tff(fact_10623_ATP_Olambda__419,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_op(Uu2),Uua),Uub),Uuc) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uu2),Uub)) ).

% ATP.lambda_419
tff(fact_10624_ATP_Olambda__420,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nm(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu2)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one)))),Uuc))) ).

% ATP.lambda_420
tff(fact_10625_ATP_Olambda__421,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_no(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu2)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one)))),Uuc))) ).

% ATP.lambda_421
tff(fact_10626_ATP_Olambda__422,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nk(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uua,Uu2)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,bit0(one)))),Uuc))) ).

% ATP.lambda_422
tff(fact_10627_ATP_Olambda__423,axiom,
    ! [Uu2: nat,Uua: nat,Uub: vEBT_VEBTi,Uuc: bool] : aa_bool_assn(aa_VEB1972364096475303967l_assn(aa_nat617759954745536040l_assn(aTP_Lamm_ba(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(treeList,Uu2)),Uub)),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uuc),aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_vebt_member,nth_VEBT_VEBT(treeList,Uu2)),Uua)))) ).

% ATP.lambda_423
tff(fact_10628_ATP_Olambda__424,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qh(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_424
tff(fact_10629_ATP_Olambda__425,axiom,
    ! [Uu2: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_oz(Uu2),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),Uuc)) ).

% ATP.lambda_425
tff(fact_10630_ATP_Olambda__426,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_oy(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu2),Uuc)) ).

% ATP.lambda_426
tff(fact_10631_ATP_Olambda__427,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_pc(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc)) ).

% ATP.lambda_427
tff(fact_10632_ATP_Olambda__428,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_pa(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu2),Uuc)) ).

% ATP.lambda_428
tff(fact_10633_ATP_Olambda__429,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_pb(Uu2),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu2),Uuc)) ).

% ATP.lambda_429
tff(fact_10634_ATP_Olambda__430,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_om(Uu2),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),Uuc)),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(Uu2,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_430
tff(fact_10635_ATP_Olambda__431,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ok(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_431
tff(fact_10636_ATP_Olambda__432,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_os(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_432
tff(fact_10637_ATP_Olambda__433,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_oo(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu2),Uuc)),minus_minus_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uu2,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_433
tff(fact_10638_ATP_Olambda__434,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_oq(Uu2),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Uu2),Uuc)),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_434
tff(fact_10639_ATP_Olambda__435,axiom,
    ! [Uu2: code_integer,Uua: nat,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_nat2351529209232473362nteger(aTP_Lamm_ou(Uu2),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Uu2),Uuc)),aa_nat_Code_integer(power_8256067586552552935nteger(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_435
tff(fact_10640_ATP_Olambda__436,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ot(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_436
tff(fact_10641_ATP_Olambda__437,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ox(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_437
tff(fact_10642_ATP_Olambda__438,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ov(Uu2),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_438
tff(fact_10643_ATP_Olambda__439,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_ow(Uu2),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Uu2),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_439
tff(fact_10644_ATP_Olambda__440,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qg(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_440
tff(fact_10645_ATP_Olambda__441,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_tk(Uu2),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(times_times_int(Uu2),Uuc)),aa_int_int(times_times_int(Uua),Uub))) ) ).

% ATP.lambda_441
tff(fact_10646_ATP_Olambda__442,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: fun_VEBT_VEBT_assn,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun437439898845168857T_bool(aa_fun2621268342929013301T_bool(aTP_Lamm_gl(Uu2),Uua),Uub),Uuc))
    <=> ( member_VEBT_VEBT(Uuc,Uu2)
        & ( aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(Uua,Uuc)),aa_VEBT_VEBT_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_442
tff(fact_10647_ATP_Olambda__443,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_gv(Uu2),Uua),Uub),Uuc))
    <=> ( member_VEBT_VEBT(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_443
tff(fact_10648_ATP_Olambda__444,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_assn,Uub: fun_complex_assn,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun1925668587588942193x_bool(aa_fun8862831692315842871x_bool(aTP_Lamm_gr(Uu2),Uua),Uub),Uuc))
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_assn_assn(times_times_assn(aa_complex_assn(Uua,Uuc)),aa_complex_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_444
tff(fact_10649_ATP_Olambda__445,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_hb(Uu2),Uua),Uub),Uuc))
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_445
tff(fact_10650_ATP_Olambda__446,axiom,
    ! [Uu2: set_real,Uua: fun_real_assn,Uub: fun_real_assn,Uuc: real] :
      ( pp(aa_real_bool(aa_fun159070158279343989l_bool(aa_fun6385136150007740861l_bool(aTP_Lamm_gj(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu2))
        & ( aa_assn_assn(times_times_assn(aa_real_assn(Uua,Uuc)),aa_real_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_446
tff(fact_10651_ATP_Olambda__447,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_gt(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_447
tff(fact_10652_ATP_Olambda__448,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1272683333798124589t_bool(aa_fun7103831392407232081t_bool(aTP_Lamm_gn(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu2))
        & ( aa_assn_assn(times_times_assn(aa_nat_assn(Uua,Uuc)),aa_nat_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_448
tff(fact_10653_ATP_Olambda__449,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_gx(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_449
tff(fact_10654_ATP_Olambda__450,axiom,
    ! [Uu2: set_int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1414614411059211637t_bool(aa_fun5523883950577104445t_bool(aTP_Lamm_gp(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu2))
        & ( aa_assn_assn(times_times_assn(aa_int_assn(Uua,Uuc)),aa_int_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_450
tff(fact_10655_ATP_Olambda__451,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_gz(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_451
tff(fact_10656_ATP_Olambda__452,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun8092769222387074655T_bool(aa_fun4024526296178165685T_bool(aTP_Lamm_hf(Uu2),Uua),Uub),Uuc))
    <=> ( member_VEBT_VEBT(Uuc,Uu2)
        & ( aa_complex_complex(plus_plus_complex(aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_452
tff(fact_10657_ATP_Olambda__453,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(aa_fun5605965666826184925T_bool(aa_fun4082248197810269365T_bool(aTP_Lamm_hp(Uu2),Uua),Uub),Uuc))
    <=> ( member_VEBT_VEBT(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_453
tff(fact_10658_ATP_Olambda__454,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_hl(Uu2),Uua),Uub),Uuc))
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_454
tff(fact_10659_ATP_Olambda__455,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_hv(Uu2),Uua),Uub),Uuc))
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_455
tff(fact_10660_ATP_Olambda__456,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_hd(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu2))
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_456
tff(fact_10661_ATP_Olambda__457,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_hn(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_real_bool(member_real(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_457
tff(fact_10662_ATP_Olambda__458,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_hh(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu2))
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_458
tff(fact_10663_ATP_Olambda__459,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_hr(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_459
tff(fact_10664_ATP_Olambda__460,axiom,
    ! [Uu2: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_hj(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu2))
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_460
tff(fact_10665_ATP_Olambda__461,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ht(Uu2),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu2))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_461
tff(fact_10666_ATP_Olambda__462,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_cj(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Uub,Uua)),aa_bool_assn(pure_assn,aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(Uuc),nth_VEBT_VEBTi(Uua,Uu2)))) ).

% ATP.lambda_462
tff(fact_10667_ATP_Olambda__463,axiom,
    ! [Uu2: nat,Uua: list_option_nat,Uub: array_option_nat,Uuc: option_nat] : aa_option_nat_assn(aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_ch(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_option_nat(Uub,Uua)),aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),nth_option_nat(Uua,Uu2)))) ).

% ATP.lambda_463
tff(fact_10668_ATP_Olambda__464,axiom,
    ! [Uu2: nat,Uua: list_bool,Uub: array_bool,Uuc: bool] : aa_bool_assn(aa_arr3483125744708554596l_assn(aa_lis2577335236791371354l_assn(aTP_Lamm_cg(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_bool(Uub,Uua)),aa_bool_assn(pure_assn,aa_bool_bool(fequal_bool(Uuc),aa_nat_bool(nth_bool(Uua),Uu2)))) ).

% ATP.lambda_464
tff(fact_10669_ATP_Olambda__465,axiom,
    ! [Uu2: nat,Uua: list_nat,Uub: array_nat,Uuc: nat] : aa_nat_assn(aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ci(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_nat(Uub,Uua)),aa_bool_assn(pure_assn,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),aa_nat_nat(nth_nat(Uua),Uu2)))) ).

% ATP.lambda_465
tff(fact_10670_ATP_Olambda__466,axiom,
    ! [Uu2: nat,Uua: list_int,Uub: array_int,Uuc: int] : aa_int_assn(aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_cf(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_int(Uub,Uua)),aa_bool_assn(pure_assn,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),aa_nat_int(nth_int(Uua),Uu2)))) ).

% ATP.lambda_466
tff(fact_10671_ATP_Olambda__467,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_fo(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_467
tff(fact_10672_ATP_Olambda__468,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fp(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_468
tff(fact_10673_ATP_Olambda__469,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_qj(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_469
tff(fact_10674_ATP_Olambda__470,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_jl(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_470
tff(fact_10675_ATP_Olambda__471,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_jn(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_471
tff(fact_10676_ATP_Olambda__472,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jj(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_472
tff(fact_10677_ATP_Olambda__473,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_jp(Uu2),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_473
tff(fact_10678_ATP_Olambda__474,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_br(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),vEBT_vebt_mint(nth_VEBT_VEBT(Uua,Uu2))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_474
tff(fact_10679_ATP_Olambda__475,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_bs(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_bool_assn(pure_assn,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,Uuc),vEBT_vebt_maxt(nth_VEBT_VEBT(Uua,Uu2))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_475
tff(fact_10680_ATP_Olambda__476,axiom,
    ! [Uu2: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(aa_fun4659777589554175228i_assn(aTP_Lamm_fy(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu2),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_476
tff(fact_10681_ATP_Olambda__477,axiom,
    ! [Uu2: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: option_nat] : aa_option_nat_assn(aa_VEB931861714549014589t_assn(aa_fun1240582285364316732t_assn(aTP_Lamm_fu(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu2),aa_option_nat_assn(aa_VEB931861714549014589t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_477
tff(fact_10682_ATP_Olambda__478,axiom,
    ! [Uu2: assn,Uua: fun_VE1323945509527881313l_assn,Uub: vEBT_VEBT,Uuc: bool] : aa_bool_assn(aa_VEB2879522196611654106l_assn(aa_fun6755658226996116224l_assn(aTP_Lamm_fs(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu2),aa_bool_assn(aa_VEB2879522196611654106l_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_478
tff(fact_10683_ATP_Olambda__479,axiom,
    ! [Uu2: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat] : aa_nat_assn(aa_VEB8640158415945897581t_assn(aa_fun7634423771582967228t_assn(aTP_Lamm_fw(Uu2),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(Uu2),aa_nat_assn(aa_VEB8640158415945897581t_assn(Uua,Uub),Uuc)) ).

% ATP.lambda_479
tff(fact_10684_ATP_Olambda__480,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lx(Uu2),Uua),Uub),Uuc) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_480
tff(fact_10685_ATP_Olambda__481,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lw(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ).

% ATP.lambda_481
tff(fact_10686_ATP_Olambda__482,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lg(Uu2),Uua),Uub),Uuc) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_482
tff(fact_10687_ATP_Olambda__483,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lf(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_483
tff(fact_10688_ATP_Olambda__484,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_tm(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(Uu2),Uuc)),aa_int_int(times_times_int(Uub),Uua)),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_484
tff(fact_10689_ATP_Olambda__485,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_to(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(times_times_int(Uu2),Uuc),aa_int_int(times_times_int(Uua),Uub))) ).

% ATP.lambda_485
tff(fact_10690_ATP_Olambda__486,axiom,
    ! [Uu2: fun_VE2561185598974572172i_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis4074100605738197896l_assn(aTP_Lamm_bf(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L1891944875198410415_VEBTi(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_486
tff(fact_10691_ATP_Olambda__487,axiom,
    ! [Uu2: fun_VE5577048379444288083T_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis183111350483865165l_assn(aTP_Lamm_be(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L7265847600308530106T_VEBT(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_487
tff(fact_10692_ATP_Olambda__488,axiom,
    ! [Uu2: fun_VE1225457145571040039t_assn,Uua: list_VEBT_VEBTi,Uub: list_nat,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis3503319847846646691l_assn(aTP_Lamm_bg(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L8930081998596925642Ti_nat(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_488
tff(fact_10693_ATP_Olambda__489,axiom,
    ! [Uu2: fun_VE5641251438918192003t_assn,Uua: list_VEBT_VEBTi,Uub: list_int,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis1781686709860263423l_assn(aTP_Lamm_bh(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L8927591528087875366Ti_int(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_489
tff(fact_10694_ATP_Olambda__490,axiom,
    ! [Uu2: fun_VE1571388752190420990T_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7769146593908314002l_assn(aTP_Lamm_bb(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L1279224858307276611T_VEBT(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_490
tff(fact_10695_ATP_Olambda__491,axiom,
    ! [Uu2: fun_VE4931415397359990460t_assn,Uua: list_VEBT_VEBT,Uub: list_nat,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis5809874432478338334l_assn(aTP_Lamm_bc(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uu2,Uua),Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_491
tff(fact_10696_ATP_Olambda__492,axiom,
    ! [Uu2: fun_VE123837653852366616t_assn,Uua: list_VEBT_VEBT,Uub: list_int,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis7389664606029993405l_assn(aa_lis4088241294491955066l_assn(aTP_Lamm_bd(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L8294436054247626077BT_int(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_492
tff(fact_10697_ATP_Olambda__493,axiom,
    ! [Uu2: fun_re560840401387337987i_assn,Uua: list_real,Uub: list_VEBT_VEBTi,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9992151435860550l_assn(aa_lis3027762689479947775l_assn(aTP_Lamm_bj(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L9060850011106065574_VEBTi(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_493
tff(fact_10698_ATP_Olambda__494,axiom,
    ! [Uu2: fun_re8050108611700907676T_assn,Uua: list_real,Uub: list_VEBT_VEBT,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis1262769258571824961l_assn(aa_lis7705788952316199574l_assn(aTP_Lamm_bi(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L4595930785310033027T_VEBT(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_494
tff(fact_10699_ATP_Olambda__495,axiom,
    ! [Uu2: fun_re574865750389657758t_assn,Uua: list_real,Uub: list_nat,Uuc: fun_bool_assn,Uud: bool] : aa_bool_assn(aa_fun923664478788943164l_assn(aa_lis9111297744016376673l_assn(aa_lis4366942040065530138l_assn(aTP_Lamm_bk(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(vEBT_L1446010312343316929al_nat(Uu2,Uua,Uub)),aa_bool_assn(Uuc,Uud)) ).

% ATP.lambda_495
tff(fact_10700_ATP_Olambda__496,axiom,
    ! [Uu2: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_rt(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rs(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))))) ).

% ATP.lambda_496
tff(fact_10701_ATP_Olambda__497,axiom,
    ! [Uu2: assn,Uua: fun_VE905575822219544865i_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_nat6700791848136694008i_assn(aa_VEB320677594755043119i_assn(aa_fun8635824850342594777i_assn(aTP_Lamm_fz(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu2),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_497
tff(fact_10702_ATP_Olambda__498,axiom,
    ! [Uu2: assn,Uua: fun_VE6334927249922962060t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_option_nat] : aa_lis1398067190886521878t_assn(aa_nat5248546159757461219t_assn(aa_VEB2926451386496479642t_assn(aa_fun7129419364150962969t_assn(aTP_Lamm_fv(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu2),aa_lis1398067190886521878t_assn(vEBT_L8010285020845282001on_nat(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_498
tff(fact_10703_ATP_Olambda__499,axiom,
    ! [Uu2: assn,Uua: fun_VE1323945509527881313l_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_bool] : aa_list_bool_assn(aa_nat2784677144853144906l_assn(aa_VEB3184745993655927913l_assn(aa_fun7303525306808986255l_assn(aTP_Lamm_ft(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu2),aa_list_bool_assn(vEBT_L5320321625355440422T_bool(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_499
tff(fact_10704_ATP_Olambda__500,axiom,
    ! [Uu2: assn,Uua: fun_VE4931415397359990460t_assn,Uub: vEBT_VEBT,Uuc: nat,Uud: list_nat] : aa_list_nat_assn(aa_nat7967224596217534867t_assn(aa_VEB8402767592425156170t_assn(aa_fun6018615163312975129t_assn(aTP_Lamm_fx(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(times_times_assn(Uu2),aa_list_nat_assn(vEBT_L8296926524756676353BT_nat(Uua,replicate_VEBT_VEBT(Uuc,Uub)),Uud)) ).

% ATP.lambda_500
tff(fact_10705_ATP_Olambda__501,axiom,
    ! [Uu2: vEBT_VEBT,Uua: real] : aa_real_VEBT_VEBT(aTP_Lamm_gc(Uu2),Uua) = Uu2 ).

% ATP.lambda_501
tff(fact_10706_ATP_Olambda__502,axiom,
    ! [Uu2: vEBT_VEBT,Uua: bool] : aa_bool_VEBT_VEBT(aTP_Lamm_gd(Uu2),Uua) = Uu2 ).

% ATP.lambda_502
tff(fact_10707_ATP_Olambda__503,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_gg(Uu2),Uua) = Uu2 ).

% ATP.lambda_503
tff(fact_10708_ATP_Olambda__504,axiom,
    ! [Uu2: vEBT_VEBT,Uua: int] : aa_int_VEBT_VEBT(aTP_Lamm_gh(Uu2),Uua) = Uu2 ).

% ATP.lambda_504
tff(fact_10709_ATP_Olambda__505,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ez(Uu2),Uua) = Uu2 ).

% ATP.lambda_505
tff(fact_10710_ATP_Olambda__506,axiom,
    ! [Uu2: real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_ga(Uu2),Uua) = Uu2 ).

% ATP.lambda_506
tff(fact_10711_ATP_Olambda__507,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fa,Uu2),Uua) = Uu2 ).

% ATP.lambda_507
tff(fact_10712_ATP_Olambda__508,axiom,
    ! [Uu2: bool,Uua: nat] :
      ( pp(aa_nat_bool(aa_bool_fun_nat_bool(aTP_Lamm_ge,Uu2),Uua))
    <=> pp(Uu2) ) ).

% ATP.lambda_508
tff(fact_10713_ATP_Olambda__509,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_nat(aTP_Lamm_gb(Uu2),Uua) = Uu2 ).

% ATP.lambda_509
tff(fact_10714_ATP_Olambda__510,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gf,Uu2),Uua) = Uu2 ).

% ATP.lambda_510
tff(fact_10715_ATP_Olambda__511,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_oi,Uu2) = Uu2 ).

% ATP.lambda_511
tff(fact_10716_ATP_Olambda__512,axiom,
    ! [Uu2: assn] : aa_assn_assn(aTP_Lamm_ao,Uu2) = Uu2 ).

% ATP.lambda_512
tff(fact_10717_ATP_Olambda__513,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ap,Uu2) = Uu2 ).

% ATP.lambda_513
tff(fact_10718_ATP_Olambda__514,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_aq,Uu2) = Uu2 ).

% ATP.lambda_514
tff(fact_10719_ATP_Olambda__515,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_ar,Uu2) = Uu2 ).

% ATP.lambda_515
tff(fact_10720_ATP_Olambda__516,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_as,Uu2) = Uu2 ).

% ATP.lambda_516
tff(fact_10721_ATP_Olambda__517,axiom,
    ! [Uu2: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_aj,Uu2) = zero_z3403309356797280102nteger ).

% ATP.lambda_517
tff(fact_10722_ATP_Olambda__518,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_ai,Uu2) = zero_zero_complex ).

% ATP.lambda_518
tff(fact_10723_ATP_Olambda__519,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ak,Uu2) = zero_zero_real ).

% ATP.lambda_519
tff(fact_10724_ATP_Olambda__520,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_al,Uu2) = zero_zero_rat ).

% ATP.lambda_520
tff(fact_10725_ATP_Olambda__521,axiom,
    ! [Uu2: nat] : aa_nat_complex(aTP_Lamm_dx,Uu2) = zero_zero_complex ).

% ATP.lambda_521
tff(fact_10726_ATP_Olambda__522,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_dy,Uu2) = zero_zero_real ).

% ATP.lambda_522
tff(fact_10727_ATP_Olambda__523,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_am,Uu2) = zero_zero_nat ).

% ATP.lambda_523
tff(fact_10728_ATP_Olambda__524,axiom,
    ! [Uu2: nat] : aa_nat_int(aTP_Lamm_dz,Uu2) = zero_zero_int ).

% ATP.lambda_524
tff(fact_10729_ATP_Olambda__525,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_an,Uu2) = zero_zero_int ).

% ATP.lambda_525
tff(fact_10730_ATP_Olambda__526,axiom,
    ! [Uu2: real] :
      ( pp(aa_real_bool(aTP_Lamm_qy,Uu2))
    <=> $false ) ).

% ATP.lambda_526
tff(fact_10731_ATP_Olambda__527,axiom,
    ! [Uu2: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_tz,Uu2))
    <=> $false ) ).

% ATP.lambda_527
tff(fact_10732_ATP_Olambda__528,axiom,
    ! [Uu2: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_uk,Uu2))
    <=> $true ) ).

% ATP.lambda_528
tff(fact_10733_ATP_Olambda__529,axiom,
    ! [Uu2: real] : aa_real_fun_nat_nat(aTP_Lamm_bl,Uu2) = suc ).

% ATP.lambda_529
tff(fact_10734_ATP_Olambda__530,axiom,
    ! [Uu2: bool] : aa_bool_fun_nat_nat(aTP_Lamm_bm,Uu2) = suc ).

% ATP.lambda_530
tff(fact_10735_ATP_Olambda__531,axiom,
    ! [Uu2: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bn,Uu2) = suc ).

% ATP.lambda_531
tff(fact_10736_ATP_Olambda__532,axiom,
    ! [Uu2: int] : aa_int_fun_nat_nat(aTP_Lamm_bo,Uu2) = suc ).

% ATP.lambda_532

% Helper facts (65)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fimplies(P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fimplies(P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fimplies(P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : if_rat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : if_rat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(fequal_bool(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(fequal_bool(X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(if_VEBT_VEBT(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X != Y )
      | pp(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( X != Y )
      | pp(aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ~ pp(aa_VEBT_VEBTi_bool(fequal_VEBT_VEBTi(X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( X != Y )
      | pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( X != Y )
      | pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(fequal_option_num,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ~ pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(fequal_option_num,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ( X != Y )
      | pp(fequal1389750275112723248_VEBTi(X,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ~ pp(fequal1389750275112723248_VEBTi(X,Y))
      | ( X = Y ) ) ).

tff(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(fFalse,X,Y) = Y ).

tff(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(fTrue,X,Y) = X ).

tff(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] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X),Y) = Y ).

tff(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] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( X != Y )
      | pp(aa_Pro4501881432599481050t_bool(fequal6010348034681699726nt_int(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ~ pp(aa_Pro4501881432599481050t_bool(fequal6010348034681699726nt_int(X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( X != Y )
      | pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ~ pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(fequal3124746952608472732at_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(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(fFalse,X,Y) = Y ).

tff(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(fTrue,X,Y) = X ).

% Conjectures (8)
tff(conj_0,hypothesis,
    tia = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),x13),x14) ).

tff(conj_1,hypothesis,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),xa)) ).

tff(conj_2,hypothesis,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).

tff(conj_3,hypothesis,
    xa != ma ).

tff(conj_4,hypothesis,
    xa != mi ).

tff(conj_5,hypothesis,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,bit0(one))))))),size_s6755466524823107622T_VEBT(treeList))) ).

tff(conj_6,hypothesis,
    size_s7982070591426661849_VEBTi(tree_is) = size_s6755466524823107622T_VEBT(treeList) ).

tff(conj_7,conjecture,
    entails(aa_assn_assn(times_times_assn(vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(treeList)),vEBT_vebt_assn_raw,treeList,tree_is)),aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(x13,tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),x14))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),x14)),snga_assn_VEBT_VEBTi(x13,tree_is_103_ATP))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,treeList),tree_is_103_ATP))) ).

%------------------------------------------------------------------------------